harmonic mean curvature flow and geometric...
TRANSCRIPT
Harmonic mean curvature flow and geometricinequalities
Haizhong Li (Tsinghua University)
Beijing Normal University, Beijing
June 5, 2019
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
This talk is based on the joint work with Ben Andrews (Australia NationalUniversity) and Yingxiang Hu (YMSC, Tsinghua University).
B. Andrews, Y. Hu and H. Li, Harmonic mean curvature flow and geometricinequalities, arXiv:1903.05903, (2019).
HMCF Haizhong Li THU 2 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
OUTLINE
1 ISOPERIMETRIC INEQUALITY IN EUCLIDEAN SPACE
2 ALEXANDROV-FENCHEL INEQUALITY IN EUCLIDEAN SPACE
3 ALEXANDROV-FENCHEL INEQUALITY IN HYPERBOLIC SPACE
4 MAIN RESULTS
HMCF Haizhong Li THU 3 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
ISOPERIMETRIC INEQUALITY IN Rn
One of the most well-known geometric inequalities for hypersurfaces in Rn
is isoperimetric inequality :
ISOPERIMETRIC INEQUALITY IN Rn
For any bounded domain Ω ⊂ Rn with boundary Σ = ∂Ω, we have
|Σ| ≥ ωn−1
(n|Ω|ωn−1
) n−1n
,
where ωn−1 is the area of the unit sphere Sn−1 ⊂ Rn. Equality holds iff Ω is ageodesic ball.
Remark. Isoperimetric inequality imposes NO convexity assumption on Σ.
HMCF Haizhong Li THU 4 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
ISOPERIMETRIC INEQUALITY IN Rn
One of the most well-known geometric inequalities for hypersurfaces in Rn
is isoperimetric inequality :
ISOPERIMETRIC INEQUALITY IN Rn
For any bounded domain Ω ⊂ Rn with boundary Σ = ∂Ω, we have
|Σ| ≥ ωn−1
(n|Ω|ωn−1
) n−1n
,
where ωn−1 is the area of the unit sphere Sn−1 ⊂ Rn. Equality holds iff Ω is ageodesic ball.
Remark. Isoperimetric inequality imposes NO convexity assumption on Σ.
HMCF Haizhong Li THU 4 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
ISOPERIMETRIC INEQUALITY IN Rn
One of the most well-known geometric inequalities for hypersurfaces in Rn
is isoperimetric inequality :
ISOPERIMETRIC INEQUALITY IN Rn
For any bounded domain Ω ⊂ Rn with boundary Σ = ∂Ω, we have
|Σ| ≥ ωn−1
(n|Ω|ωn−1
) n−1n
,
where ωn−1 is the area of the unit sphere Sn−1 ⊂ Rn. Equality holds iff Ω is ageodesic ball.
Remark. Isoperimetric inequality imposes NO convexity assumption on Σ.
HMCF Haizhong Li THU 4 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Rn
Let κ = (κ1, · · · , κn−1) be the principal curvatures of a hypersurfaceΣ ⊂ Rn.
The (normalized) m-th mean curvature pm of Σ is
p0 = 1, pm(κ) =1
Cmn−1
∑1≤i1<i2<···<im≤n−1
κi1 · · ·κim .
The natural generalization of isoperimetric inequality in Rn isAlexandrov-Fenchel inequality :
ALEXANDROV-FENCHEL INEQUALITY IN Rn
For any closed convex hypersurface Σ ⊂ Rn, we have∫Σ
pm(κ)dµ ≥ ωm−l
n−1−ln−1
(∫Σ
pl (κ)dµ) n−1−m
n−1−l
, 0 ≤ l < m ≤ n − 1.
Equality holds iff Σ is a geodesic sphere.
HMCF Haizhong Li THU 5 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Rn
Let κ = (κ1, · · · , κn−1) be the principal curvatures of a hypersurfaceΣ ⊂ Rn.The (normalized) m-th mean curvature pm of Σ is
p0 = 1, pm(κ) =1
Cmn−1
∑1≤i1<i2<···<im≤n−1
κi1 · · ·κim .
The natural generalization of isoperimetric inequality in Rn isAlexandrov-Fenchel inequality :
ALEXANDROV-FENCHEL INEQUALITY IN Rn
For any closed convex hypersurface Σ ⊂ Rn, we have∫Σ
pm(κ)dµ ≥ ωm−l
n−1−ln−1
(∫Σ
pl (κ)dµ) n−1−m
n−1−l
, 0 ≤ l < m ≤ n − 1.
Equality holds iff Σ is a geodesic sphere.
HMCF Haizhong Li THU 5 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Rn
Let κ = (κ1, · · · , κn−1) be the principal curvatures of a hypersurfaceΣ ⊂ Rn.The (normalized) m-th mean curvature pm of Σ is
p0 = 1, pm(κ) =1
Cmn−1
∑1≤i1<i2<···<im≤n−1
κi1 · · ·κim .
The natural generalization of isoperimetric inequality in Rn isAlexandrov-Fenchel inequality :
ALEXANDROV-FENCHEL INEQUALITY IN Rn
For any closed convex hypersurface Σ ⊂ Rn, we have∫Σ
pm(κ)dµ ≥ ωm−l
n−1−ln−1
(∫Σ
pl (κ)dµ) n−1−m
n−1−l
, 0 ≤ l < m ≤ n − 1.
Equality holds iff Σ is a geodesic sphere.
HMCF Haizhong Li THU 5 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Rn
Let κ = (κ1, · · · , κn−1) be the principal curvatures of a hypersurfaceΣ ⊂ Rn.The (normalized) m-th mean curvature pm of Σ is
p0 = 1, pm(κ) =1
Cmn−1
∑1≤i1<i2<···<im≤n−1
κi1 · · ·κim .
The natural generalization of isoperimetric inequality in Rn isAlexandrov-Fenchel inequality :
ALEXANDROV-FENCHEL INEQUALITY IN Rn
For any closed convex hypersurface Σ ⊂ Rn, we have∫Σ
pm(κ)dµ ≥ ωm−l
n−1−ln−1
(∫Σ
pl (κ)dµ) n−1−m
n−1−l
, 0 ≤ l < m ≤ n − 1.
Equality holds iff Σ is a geodesic sphere.
HMCF Haizhong Li THU 5 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Rn
Taking l = 0, for any closed convex hypersurface Σ ⊂ Rn, we have∫Σ
pm(κ)dµ ≥ ωn−1
(|Σ|ωn−1
) n−1−mn−l
, 1 ≤ m ≤ n − 1.
Equality holds iff Σ is a geodesic sphere.
Isoperimetric inequality imposes NO convexity assumption, so it is naturalto weaken convexity assumption for AF inequality.
A hypersurface Σ is called starshaped, if 〈∂r , ν〉 > 0 on Σ, where ν isthe unit outward normal of Σ and ∂r is the radial vector, respectively.A hypersurface Σ is called m-convex, if pi (κ) > 0 for i = 1, · · · ,meverywhere on Σ.
HMCF Haizhong Li THU 6 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Rn
Taking l = 0, for any closed convex hypersurface Σ ⊂ Rn, we have∫Σ
pm(κ)dµ ≥ ωn−1
(|Σ|ωn−1
) n−1−mn−l
, 1 ≤ m ≤ n − 1.
Equality holds iff Σ is a geodesic sphere.
Isoperimetric inequality imposes NO convexity assumption, so it is naturalto weaken convexity assumption for AF inequality.
A hypersurface Σ is called starshaped, if 〈∂r , ν〉 > 0 on Σ, where ν isthe unit outward normal of Σ and ∂r is the radial vector, respectively.A hypersurface Σ is called m-convex, if pi (κ) > 0 for i = 1, · · · ,meverywhere on Σ.
HMCF Haizhong Li THU 6 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Rn
Taking l = 0, for any closed convex hypersurface Σ ⊂ Rn, we have∫Σ
pm(κ)dµ ≥ ωn−1
(|Σ|ωn−1
) n−1−mn−l
, 1 ≤ m ≤ n − 1.
Equality holds iff Σ is a geodesic sphere.
Isoperimetric inequality imposes NO convexity assumption, so it is naturalto weaken convexity assumption for AF inequality.
A hypersurface Σ is called starshaped, if 〈∂r , ν〉 > 0 on Σ, where ν isthe unit outward normal of Σ and ∂r is the radial vector, respectively.
A hypersurface Σ is called m-convex, if pi (κ) > 0 for i = 1, · · · ,meverywhere on Σ.
HMCF Haizhong Li THU 6 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Rn
Taking l = 0, for any closed convex hypersurface Σ ⊂ Rn, we have∫Σ
pm(κ)dµ ≥ ωn−1
(|Σ|ωn−1
) n−1−mn−l
, 1 ≤ m ≤ n − 1.
Equality holds iff Σ is a geodesic sphere.
Isoperimetric inequality imposes NO convexity assumption, so it is naturalto weaken convexity assumption for AF inequality.
A hypersurface Σ is called starshaped, if 〈∂r , ν〉 > 0 on Σ, where ν isthe unit outward normal of Σ and ∂r is the radial vector, respectively.A hypersurface Σ is called m-convex, if pi (κ) > 0 for i = 1, · · · ,meverywhere on Σ.
HMCF Haizhong Li THU 6 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Rn
By using the smooth convergence of inverse curvature flows in Rn by C.Gerhardt (1990) and J. Urbas (1990), Pengfei Guan and Junfang Li provedthat
THEOREM, P. GUAN AND J. LI, 2009, ADV. MATH.
If hypersurface Σ ⊂ Rn(n ≥ 3) is star-shaped and m-convex, then∫Σ
pm(κ)dµ ≥ ωn−1
(|Σ|ωn−1
) n−1−mn−l
, 1 ≤ m ≤ n − 1.
Equality holds iff Σ is a geodesic sphere.
HMCF Haizhong Li THU 7 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Rn
By using the smooth convergence of inverse curvature flows in Rn by C.Gerhardt (1990) and J. Urbas (1990), Pengfei Guan and Junfang Li provedthat
THEOREM, P. GUAN AND J. LI, 2009, ADV. MATH.
If hypersurface Σ ⊂ Rn(n ≥ 3) is star-shaped and m-convex, then∫Σ
pm(κ)dµ ≥ ωn−1
(|Σ|ωn−1
) n−1−mn−l
, 1 ≤ m ≤ n − 1.
Equality holds iff Σ is a geodesic sphere.
HMCF Haizhong Li THU 7 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
ISOPERIMETRIC INEQUALITY IN Hn
For n = 2, for a simple closed curve γ in H2, we have
L2 ≥ 4πV + V 2,
where L is the length of γ and V is the volume of the domain enclosed by γ.Moreover, Equality holds if and only if γ is a circle.
For n ≥ 3, the isoperimetric inequality in Hn was proved by E. Schmidt(1940), but the explicit expression is rare.
HMCF Haizhong Li THU 8 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
ISOPERIMETRIC INEQUALITY IN Hn
For n = 2, for a simple closed curve γ in H2, we have
L2 ≥ 4πV + V 2,
where L is the length of γ and V is the volume of the domain enclosed by γ.Moreover, Equality holds if and only if γ is a circle.
For n ≥ 3, the isoperimetric inequality in Hn was proved by E. Schmidt(1940), but the explicit expression is rare.
HMCF Haizhong Li THU 8 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
In 2011, C. Gerhardt established the smooth convergence of inversecurvature flows in hyperbolic space.
PROBLEM
Establish an analogue of Alexandrov-Fenchel inequality in Hn.
Joint with Yong Wei and Changwei Xiong, we apply the inverse curvatureflow to obtain
THEOREM, L.-WEI-XIONG, 2014, ADV. MATH
If hypersurface Σ ⊂ Hn(n ≥ 3) is starshaped and 2-convex, then∫Σ
p2 ≥ ωn−1
[(|Σ|ωn−1
)+
(|Σ|ωn−1
) n−3n−1].
Equality holds iff Σ is a geodesic sphere.
HMCF Haizhong Li THU 9 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
In 2011, C. Gerhardt established the smooth convergence of inversecurvature flows in hyperbolic space.
PROBLEM
Establish an analogue of Alexandrov-Fenchel inequality in Hn.
Joint with Yong Wei and Changwei Xiong, we apply the inverse curvatureflow to obtain
THEOREM, L.-WEI-XIONG, 2014, ADV. MATH
If hypersurface Σ ⊂ Hn(n ≥ 3) is starshaped and 2-convex, then∫Σ
p2 ≥ ωn−1
[(|Σ|ωn−1
)+
(|Σ|ωn−1
) n−3n−1].
Equality holds iff Σ is a geodesic sphere.
HMCF Haizhong Li THU 9 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
In 2011, C. Gerhardt established the smooth convergence of inversecurvature flows in hyperbolic space.
PROBLEM
Establish an analogue of Alexandrov-Fenchel inequality in Hn.
Joint with Yong Wei and Changwei Xiong, we apply the inverse curvatureflow to obtain
THEOREM, L.-WEI-XIONG, 2014, ADV. MATH
If hypersurface Σ ⊂ Hn(n ≥ 3) is starshaped and 2-convex, then∫Σ
p2 ≥ ωn−1
[(|Σ|ωn−1
)+
(|Σ|ωn−1
) n−3n−1].
Equality holds iff Σ is a geodesic sphere.
HMCF Haizhong Li THU 9 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof of Li-Wei-Xiong’s inequality consists of four steps:
The convergence result of inverse curvature flows ∂tX = p1p2ν in Hn by
C. Gerhardt;The preservance of 2-convexity along this flow;The monotonicity of
Q(t) = |Σt |−n−3n−1
∫Σ
(p2 − 1),
where the 2-convexity of Σt plays an essential role;The Sobolev inequality by W. Beckner (1993), which is used to show
limt→∞Q(t) ≥ ω2
n−1n−1.
HMCF Haizhong Li THU 10 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof of Li-Wei-Xiong’s inequality consists of four steps:The convergence result of inverse curvature flows ∂tX = p1
p2ν in Hn by
C. Gerhardt;
The preservance of 2-convexity along this flow;The monotonicity of
Q(t) = |Σt |−n−3n−1
∫Σ
(p2 − 1),
where the 2-convexity of Σt plays an essential role;The Sobolev inequality by W. Beckner (1993), which is used to show
limt→∞Q(t) ≥ ω2
n−1n−1.
HMCF Haizhong Li THU 10 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof of Li-Wei-Xiong’s inequality consists of four steps:The convergence result of inverse curvature flows ∂tX = p1
p2ν in Hn by
C. Gerhardt;The preservance of 2-convexity along this flow;
The monotonicity of
Q(t) = |Σt |−n−3n−1
∫Σ
(p2 − 1),
where the 2-convexity of Σt plays an essential role;The Sobolev inequality by W. Beckner (1993), which is used to show
limt→∞Q(t) ≥ ω2
n−1n−1.
HMCF Haizhong Li THU 10 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof of Li-Wei-Xiong’s inequality consists of four steps:The convergence result of inverse curvature flows ∂tX = p1
p2ν in Hn by
C. Gerhardt;The preservance of 2-convexity along this flow;The monotonicity of
Q(t) = |Σt |−n−3n−1
∫Σ
(p2 − 1),
where the 2-convexity of Σt plays an essential role;
The Sobolev inequality by W. Beckner (1993), which is used to show
limt→∞Q(t) ≥ ω2
n−1n−1.
HMCF Haizhong Li THU 10 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof of Li-Wei-Xiong’s inequality consists of four steps:The convergence result of inverse curvature flows ∂tX = p1
p2ν in Hn by
C. Gerhardt;The preservance of 2-convexity along this flow;The monotonicity of
Q(t) = |Σt |−n−3n−1
∫Σ
(p2 − 1),
where the 2-convexity of Σt plays an essential role;The Sobolev inequality by W. Beckner (1993), which is used to show
limt→∞Q(t) ≥ ω2
n−1n−1.
HMCF Haizhong Li THU 10 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
Arising naturally from Li-Wei-Xiong’s inequality, the following Conjecture isstill open:
CONJECTURE
Let 1 ≤ k ≤ n − 1. Any starshaped and k-convex hypersurface Σ ⊂ Hn
satisfies ∫Σ
pk ≥ ωn−1
[(|Σ|ωn−1
) 2k
+
(|Σ|ωn−1
) 2k
n−1−kn−1
] k2
.
Equality holds iff Σ is a geodesic sphere.
Remark.This verifies the Conjecture for the case k = 2.With the Li-Wei-Xiong’s inequality and a result of Xu Cheng andDetang Zhou, the Conjecture for k = 1 holds for hypersurfaces withnonnegative Ricci curvature in Hn.
HMCF Haizhong Li THU 11 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
Arising naturally from Li-Wei-Xiong’s inequality, the following Conjecture isstill open:
CONJECTURE
Let 1 ≤ k ≤ n − 1. Any starshaped and k-convex hypersurface Σ ⊂ Hn
satisfies ∫Σ
pk ≥ ωn−1
[(|Σ|ωn−1
) 2k
+
(|Σ|ωn−1
) 2k
n−1−kn−1
] k2
.
Equality holds iff Σ is a geodesic sphere.
Remark.This verifies the Conjecture for the case k = 2.With the Li-Wei-Xiong’s inequality and a result of Xu Cheng andDetang Zhou, the Conjecture for k = 1 holds for hypersurfaces withnonnegative Ricci curvature in Hn.
HMCF Haizhong Li THU 11 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
Arising naturally from Li-Wei-Xiong’s inequality, the following Conjecture isstill open:
CONJECTURE
Let 1 ≤ k ≤ n − 1. Any starshaped and k-convex hypersurface Σ ⊂ Hn
satisfies ∫Σ
pk ≥ ωn−1
[(|Σ|ωn−1
) 2k
+
(|Σ|ωn−1
) 2k
n−1−kn−1
] k2
.
Equality holds iff Σ is a geodesic sphere.
Remark.This verifies the Conjecture for the case k = 2.
With the Li-Wei-Xiong’s inequality and a result of Xu Cheng andDetang Zhou, the Conjecture for k = 1 holds for hypersurfaces withnonnegative Ricci curvature in Hn.
HMCF Haizhong Li THU 11 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
Arising naturally from Li-Wei-Xiong’s inequality, the following Conjecture isstill open:
CONJECTURE
Let 1 ≤ k ≤ n − 1. Any starshaped and k-convex hypersurface Σ ⊂ Hn
satisfies ∫Σ
pk ≥ ωn−1
[(|Σ|ωn−1
) 2k
+
(|Σ|ωn−1
) 2k
n−1−kn−1
] k2
.
Equality holds iff Σ is a geodesic sphere.
Remark.This verifies the Conjecture for the case k = 2.With the Li-Wei-Xiong’s inequality and a result of Xu Cheng andDetang Zhou, the Conjecture for k = 1 holds for hypersurfaces withnonnegative Ricci curvature in Hn.
HMCF Haizhong Li THU 11 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
CONVEXITY IN HYPERBOLIC SPACE
To state the recent developments on this Conjecture, we recall the variousconvexity for hypersurfaces in hyperbolic space.
Under stronger convexity assumptions, the inequality in Conjecture isTRUE.
Different from a hypersurface in Rn, there are four different kinds ofconvexity (in strictly ascending order) for a hypersurface (Σ,g) in Hn:
(strictly) convex if κi > 0 for i = 1, · · · ,n − 1;
nonnegative Ricci curvature if κi
(∑j 6=i κj
)≥ n − 2 for i = 1, · · · ,n − 1;
nonnegative sectional curvature if κiκj ≥ 1 for 1 ≤ i < j ≤ n − 1;horospherical convex (h-convex) if κi ≥ 1 for i = 1, · · · ,n − 1.
Remark. The strict convexity is equivalent to (n − 1)-convex andstarshaped, and hence all these convexity conditions are stronger thanm-convex and starshaped.
HMCF Haizhong Li THU 12 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
CONVEXITY IN HYPERBOLIC SPACE
To state the recent developments on this Conjecture, we recall the variousconvexity for hypersurfaces in hyperbolic space.
Under stronger convexity assumptions, the inequality in Conjecture isTRUE.
Different from a hypersurface in Rn, there are four different kinds ofconvexity (in strictly ascending order) for a hypersurface (Σ,g) in Hn:
(strictly) convex if κi > 0 for i = 1, · · · ,n − 1;
nonnegative Ricci curvature if κi
(∑j 6=i κj
)≥ n − 2 for i = 1, · · · ,n − 1;
nonnegative sectional curvature if κiκj ≥ 1 for 1 ≤ i < j ≤ n − 1;horospherical convex (h-convex) if κi ≥ 1 for i = 1, · · · ,n − 1.
Remark. The strict convexity is equivalent to (n − 1)-convex andstarshaped, and hence all these convexity conditions are stronger thanm-convex and starshaped.
HMCF Haizhong Li THU 12 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
CONVEXITY IN HYPERBOLIC SPACE
To state the recent developments on this Conjecture, we recall the variousconvexity for hypersurfaces in hyperbolic space.
Under stronger convexity assumptions, the inequality in Conjecture isTRUE.
Different from a hypersurface in Rn, there are four different kinds ofconvexity (in strictly ascending order) for a hypersurface (Σ,g) in Hn:
(strictly) convex if κi > 0 for i = 1, · · · ,n − 1;
nonnegative Ricci curvature if κi
(∑j 6=i κj
)≥ n − 2 for i = 1, · · · ,n − 1;
nonnegative sectional curvature if κiκj ≥ 1 for 1 ≤ i < j ≤ n − 1;horospherical convex (h-convex) if κi ≥ 1 for i = 1, · · · ,n − 1.
Remark. The strict convexity is equivalent to (n − 1)-convex andstarshaped, and hence all these convexity conditions are stronger thanm-convex and starshaped.
HMCF Haizhong Li THU 12 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
CONVEXITY IN HYPERBOLIC SPACE
To state the recent developments on this Conjecture, we recall the variousconvexity for hypersurfaces in hyperbolic space.
Under stronger convexity assumptions, the inequality in Conjecture isTRUE.
Different from a hypersurface in Rn, there are four different kinds ofconvexity (in strictly ascending order) for a hypersurface (Σ,g) in Hn:
(strictly) convex if κi > 0 for i = 1, · · · ,n − 1;
nonnegative Ricci curvature if κi
(∑j 6=i κj
)≥ n − 2 for i = 1, · · · ,n − 1;
nonnegative sectional curvature if κiκj ≥ 1 for 1 ≤ i < j ≤ n − 1;horospherical convex (h-convex) if κi ≥ 1 for i = 1, · · · ,n − 1.
Remark. The strict convexity is equivalent to (n − 1)-convex andstarshaped, and hence all these convexity conditions are stronger thanm-convex and starshaped.
HMCF Haizhong Li THU 12 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
CONVEXITY IN HYPERBOLIC SPACE
To state the recent developments on this Conjecture, we recall the variousconvexity for hypersurfaces in hyperbolic space.
Under stronger convexity assumptions, the inequality in Conjecture isTRUE.
Different from a hypersurface in Rn, there are four different kinds ofconvexity (in strictly ascending order) for a hypersurface (Σ,g) in Hn:
(strictly) convex if κi > 0 for i = 1, · · · ,n − 1;
nonnegative Ricci curvature if κi
(∑j 6=i κj
)≥ n − 2 for i = 1, · · · ,n − 1;
nonnegative sectional curvature if κiκj ≥ 1 for 1 ≤ i < j ≤ n − 1;horospherical convex (h-convex) if κi ≥ 1 for i = 1, · · · ,n − 1.
Remark. The strict convexity is equivalent to (n − 1)-convex andstarshaped, and hence all these convexity conditions are stronger thanm-convex and starshaped.
HMCF Haizhong Li THU 12 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
In 2014, Yuxin Ge, Guofang Wang and Jie Wu investigated the k -thGauss-Bonnet curvature Lk on hypersurface (Σ,g) in Hn,
Lk :=12k δ
i1 i2···i2k−1 i2kj1 j2···j2k−1 j2k
Ri1 i2j1 j2 · · ·Ri2k−1 i2k
j2k−1 j2k ,
where Rijkl is the Riemannian curvature tensor in the local coordinates w.r.t.
the metric g, and the generalized Kronecker delta is defined by
δj1 j2···jri1 i2···ir = det
δj1
i1 δj2i1 · · · δjr
i1δj1
i2 δj2i2 · · · δjr
i2...
......
...δj1
ir δj2ir · · · δjr
ir
.
HMCF Haizhong Li THU 13 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
By using the inverse curvature flows in Hn, they proved an optimalSobolev-type inequality for h-convex hypersurfaces in Hn:
THEOREM, GE-WANG-WU, 2014, J. DIFF. GEOM.
Let 2 ≤ 2k < n − 1. Any h-convex hypersurface (Σ,g) ⊂ Hn satisfies∫Σ
Lk dµ ≥ C2kn−1(2k)!ω
2kn−1n−1|Σ|
n−1−2kn−1 .
Equality holds iff Σ is a geodesic sphere.
Remark. For any hypersurface (Σ,g) in Hn, the Gauss-Bonnet curvature Lkcan be expressed by
Lk = C2kn−1(2k)!
k∑j=0
(−1)jC jk p2k−2j .
For k = 1, the above inequality coincides with Li-Wei-Xiong’s inequality.
HMCF Haizhong Li THU 14 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
By using the inverse curvature flows in Hn, they proved an optimalSobolev-type inequality for h-convex hypersurfaces in Hn:
THEOREM, GE-WANG-WU, 2014, J. DIFF. GEOM.
Let 2 ≤ 2k < n − 1. Any h-convex hypersurface (Σ,g) ⊂ Hn satisfies∫Σ
Lk dµ ≥ C2kn−1(2k)!ω
2kn−1n−1|Σ|
n−1−2kn−1 .
Equality holds iff Σ is a geodesic sphere.
Remark. For any hypersurface (Σ,g) in Hn, the Gauss-Bonnet curvature Lkcan be expressed by
Lk = C2kn−1(2k)!
k∑j=0
(−1)jC jk p2k−2j .
For k = 1, the above inequality coincides with Li-Wei-Xiong’s inequality.
HMCF Haizhong Li THU 14 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
By using the inverse curvature flows in Hn, they proved an optimalSobolev-type inequality for h-convex hypersurfaces in Hn:
THEOREM, GE-WANG-WU, 2014, J. DIFF. GEOM.
Let 2 ≤ 2k < n − 1. Any h-convex hypersurface (Σ,g) ⊂ Hn satisfies∫Σ
Lk dµ ≥ C2kn−1(2k)!ω
2kn−1n−1|Σ|
n−1−2kn−1 .
Equality holds iff Σ is a geodesic sphere.
Remark. For any hypersurface (Σ,g) in Hn, the Gauss-Bonnet curvature Lkcan be expressed by
Lk = C2kn−1(2k)!
k∑j=0
(−1)jC jk p2k−2j .
For k = 1, the above inequality coincides with Li-Wei-Xiong’s inequality.HMCF Haizhong Li THU 14 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof of Ge-Wang-Wu’s inequality consists of four ingredients:
The convergence result of inverse curvature flows ∂tX =p2k−1
p2kν in Hn
by C. Gerhardt;The preservance of h-convexity along the inverse curvature flows;The monotonicity of
Q(t) = |Σt |−n−1−2k
n−1
∫Σ
Lk ,
where the h-convexity of Σ plays an essential role;The generalized Sobolev inequality by P. Guan and G. Wang (2003),
which is used to show limt→∞Q(t) ≥ C2kn−1(2k)!ω
2kn−1n−1.
HMCF Haizhong Li THU 15 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof of Ge-Wang-Wu’s inequality consists of four ingredients:The convergence result of inverse curvature flows ∂tX =
p2k−1p2k
ν in Hn
by C. Gerhardt;
The preservance of h-convexity along the inverse curvature flows;The monotonicity of
Q(t) = |Σt |−n−1−2k
n−1
∫Σ
Lk ,
where the h-convexity of Σ plays an essential role;The generalized Sobolev inequality by P. Guan and G. Wang (2003),
which is used to show limt→∞Q(t) ≥ C2kn−1(2k)!ω
2kn−1n−1.
HMCF Haizhong Li THU 15 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof of Ge-Wang-Wu’s inequality consists of four ingredients:The convergence result of inverse curvature flows ∂tX =
p2k−1p2k
ν in Hn
by C. Gerhardt;The preservance of h-convexity along the inverse curvature flows;
The monotonicity of
Q(t) = |Σt |−n−1−2k
n−1
∫Σ
Lk ,
where the h-convexity of Σ plays an essential role;The generalized Sobolev inequality by P. Guan and G. Wang (2003),
which is used to show limt→∞Q(t) ≥ C2kn−1(2k)!ω
2kn−1n−1.
HMCF Haizhong Li THU 15 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof of Ge-Wang-Wu’s inequality consists of four ingredients:The convergence result of inverse curvature flows ∂tX =
p2k−1p2k
ν in Hn
by C. Gerhardt;The preservance of h-convexity along the inverse curvature flows;The monotonicity of
Q(t) = |Σt |−n−1−2k
n−1
∫Σ
Lk ,
where the h-convexity of Σ plays an essential role;
The generalized Sobolev inequality by P. Guan and G. Wang (2003),
which is used to show limt→∞Q(t) ≥ C2kn−1(2k)!ω
2kn−1n−1.
HMCF Haizhong Li THU 15 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof of Ge-Wang-Wu’s inequality consists of four ingredients:The convergence result of inverse curvature flows ∂tX =
p2k−1p2k
ν in Hn
by C. Gerhardt;The preservance of h-convexity along the inverse curvature flows;The monotonicity of
Q(t) = |Σt |−n−1−2k
n−1
∫Σ
Lk ,
where the h-convexity of Σ plays an essential role;The generalized Sobolev inequality by P. Guan and G. Wang (2003),
which is used to show limt→∞Q(t) ≥ C2kn−1(2k)!ω
2kn−1n−1.
HMCF Haizhong Li THU 15 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The importance of this inequality is that it can be viewed as the bricks ofother geometric inequalities. To observe this, we have∫
Σ
p2k =1
(2k)!C2kn−1
k∑i=0
C ik
∫Σ
Li .
They proved AF inequality for curvature integrals in Hn:
THEOREM, GE-WANG-WU, 2014, J. DIFF. GEOM.
Let 2 ≤ 2k ≤ n − 1. Any h-convex hypersurface (Σ,g) ⊂ Hn satisfies
∫Σ
p2k ≥ ωn−1
[(|Σ|ωn−1
) 1k
+
(|Σ|ωn−1
) 1k
n−1−2kn−1
]k
,
Equality holds iff Σ is a geodesic sphere.
HMCF Haizhong Li THU 16 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The importance of this inequality is that it can be viewed as the bricks ofother geometric inequalities. To observe this, we have∫
Σ
p2k =1
(2k)!C2kn−1
k∑i=0
C ik
∫Σ
Li .
They proved AF inequality for curvature integrals in Hn:
THEOREM, GE-WANG-WU, 2014, J. DIFF. GEOM.
Let 2 ≤ 2k ≤ n − 1. Any h-convex hypersurface (Σ,g) ⊂ Hn satisfies
∫Σ
p2k ≥ ωn−1
[(|Σ|ωn−1
) 1k
+
(|Σ|ωn−1
) 1k
n−1−2kn−1
]k
,
Equality holds iff Σ is a geodesic sphere.
HMCF Haizhong Li THU 16 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The importance of this inequality is that it can be viewed as the bricks ofother geometric inequalities. To observe this, we have∫
Σ
p2k =1
(2k)!C2kn−1
k∑i=0
C ik
∫Σ
Li .
They proved AF inequality for curvature integrals in Hn:
THEOREM, GE-WANG-WU, 2014, J. DIFF. GEOM.
Let 2 ≤ 2k ≤ n − 1. Any h-convex hypersurface (Σ,g) ⊂ Hn satisfies
∫Σ
p2k ≥ ωn−1
[(|Σ|ωn−1
) 1k
+
(|Σ|ωn−1
) 1k
n−1−2kn−1
]k
,
Equality holds iff Σ is a geodesic sphere.
HMCF Haizhong Li THU 16 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
In a recent work joint with Yingxiang Hu, all Ge-Wang-Wu’s inequalitieshave been extended to hypersurfaces with nonnegative sectional curvaturein hyperbolic space.
In particular, we prove
THEOREM, HU-L., 2019, CALC. VAR.
Let 2 ≤ 2k ≤ n − 1. Any hypersurface (Σ,g) ⊂ Hn with nonnegativesectional curvature satisfies
∫Σ
p2k ≥ ωn−1
[(|Σ|ωn−1
) 1k
+
(|Σ|ωn−1
) 1k
n−1−2kn−1
]k
,
Equality holds iff Σ is a geodesic sphere.
Remark. The Conjecture for k = 2m (2 ≤ 2m ≤ n − 1) holds forhypersurfaces with nonnegative sectional curvature in Hn.
HMCF Haizhong Li THU 17 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
In a recent work joint with Yingxiang Hu, all Ge-Wang-Wu’s inequalitieshave been extended to hypersurfaces with nonnegative sectional curvaturein hyperbolic space. In particular, we prove
THEOREM, HU-L., 2019, CALC. VAR.
Let 2 ≤ 2k ≤ n − 1. Any hypersurface (Σ,g) ⊂ Hn with nonnegativesectional curvature satisfies
∫Σ
p2k ≥ ωn−1
[(|Σ|ωn−1
) 1k
+
(|Σ|ωn−1
) 1k
n−1−2kn−1
]k
,
Equality holds iff Σ is a geodesic sphere.
Remark. The Conjecture for k = 2m (2 ≤ 2m ≤ n − 1) holds forhypersurfaces with nonnegative sectional curvature in Hn.
HMCF Haizhong Li THU 17 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
In a recent work joint with Yingxiang Hu, all Ge-Wang-Wu’s inequalitieshave been extended to hypersurfaces with nonnegative sectional curvaturein hyperbolic space. In particular, we prove
THEOREM, HU-L., 2019, CALC. VAR.
Let 2 ≤ 2k ≤ n − 1. Any hypersurface (Σ,g) ⊂ Hn with nonnegativesectional curvature satisfies
∫Σ
p2k ≥ ωn−1
[(|Σ|ωn−1
) 1k
+
(|Σ|ωn−1
) 1k
n−1−2kn−1
]k
,
Equality holds iff Σ is a geodesic sphere.
Remark. The Conjecture for k = 2m (2 ≤ 2m ≤ n − 1) holds forhypersurfaces with nonnegative sectional curvature in Hn.
HMCF Haizhong Li THU 17 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof consists of four ingredients:
The convergence result of inverse mean curvature flow ∂tX = 1H ν in Hn
by C. Gerhardt;The preservance of nonnegative sectional curvature along the IMCF,which is inspired by the recent work of Andrews-Chen-Wei on volumepreserving flows in hyperbolic space;The monotonicity of
Q(t) = |Σt |−n−1−2k
n−1
∫Σ
Lk ,
where the nonnegative sectional curvature of Σ plays an essential role;This has already been observed by Ge-Wang-Wu;The analysis of limt→∞Q(t) is the same as Ge-Wang-Wu.
HMCF Haizhong Li THU 18 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof consists of four ingredients:The convergence result of inverse mean curvature flow ∂tX = 1
H ν in Hn
by C. Gerhardt;
The preservance of nonnegative sectional curvature along the IMCF,which is inspired by the recent work of Andrews-Chen-Wei on volumepreserving flows in hyperbolic space;The monotonicity of
Q(t) = |Σt |−n−1−2k
n−1
∫Σ
Lk ,
where the nonnegative sectional curvature of Σ plays an essential role;This has already been observed by Ge-Wang-Wu;The analysis of limt→∞Q(t) is the same as Ge-Wang-Wu.
HMCF Haizhong Li THU 18 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof consists of four ingredients:The convergence result of inverse mean curvature flow ∂tX = 1
H ν in Hn
by C. Gerhardt;The preservance of nonnegative sectional curvature along the IMCF,which is inspired by the recent work of Andrews-Chen-Wei on volumepreserving flows in hyperbolic space;
The monotonicity of
Q(t) = |Σt |−n−1−2k
n−1
∫Σ
Lk ,
where the nonnegative sectional curvature of Σ plays an essential role;This has already been observed by Ge-Wang-Wu;The analysis of limt→∞Q(t) is the same as Ge-Wang-Wu.
HMCF Haizhong Li THU 18 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof consists of four ingredients:The convergence result of inverse mean curvature flow ∂tX = 1
H ν in Hn
by C. Gerhardt;The preservance of nonnegative sectional curvature along the IMCF,which is inspired by the recent work of Andrews-Chen-Wei on volumepreserving flows in hyperbolic space;The monotonicity of
Q(t) = |Σt |−n−1−2k
n−1
∫Σ
Lk ,
where the nonnegative sectional curvature of Σ plays an essential role;This has already been observed by Ge-Wang-Wu;
The analysis of limt→∞Q(t) is the same as Ge-Wang-Wu.
HMCF Haizhong Li THU 18 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
The proof consists of four ingredients:The convergence result of inverse mean curvature flow ∂tX = 1
H ν in Hn
by C. Gerhardt;The preservance of nonnegative sectional curvature along the IMCF,which is inspired by the recent work of Andrews-Chen-Wei on volumepreserving flows in hyperbolic space;The monotonicity of
Q(t) = |Σt |−n−1−2k
n−1
∫Σ
Lk ,
where the nonnegative sectional curvature of Σ plays an essential role;This has already been observed by Ge-Wang-Wu;The analysis of limt→∞Q(t) is the same as Ge-Wang-Wu.
HMCF Haizhong Li THU 18 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
In 2014, Guofang Wang and Chao Xia used the quermassintegralpreserving curvature flows to prove AF inequality for curvature integrals inHn:
THEOREM, WANG-XIA, 2014, ADV. MATH.
Let 1 ≤ k ≤ n − 1. If Ω ⊂ Hn is a bounded domain with Σ = ∂Ω h-convex,then ∫
Σ
pk ≥ ωn−1
[(|Σ|ωn−1
) 2k
+
(|Σ|ωn−1
) 2k
n−1−kn−1
] k2
,
Equality holds iff Σ is a geodesic sphere.
Remark. The Conjecture for 1 ≤ k ≤ n − 1 holds for h-convexhypersurfaces in Hn.
HMCF Haizhong Li THU 19 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
In 2014, Guofang Wang and Chao Xia used the quermassintegralpreserving curvature flows to prove AF inequality for curvature integrals inHn:
THEOREM, WANG-XIA, 2014, ADV. MATH.
Let 1 ≤ k ≤ n − 1. If Ω ⊂ Hn is a bounded domain with Σ = ∂Ω h-convex,then ∫
Σ
pk ≥ ωn−1
[(|Σ|ωn−1
) 2k
+
(|Σ|ωn−1
) 2k
n−1−kn−1
] k2
,
Equality holds iff Σ is a geodesic sphere.
Remark. The Conjecture for 1 ≤ k ≤ n − 1 holds for h-convexhypersurfaces in Hn.
HMCF Haizhong Li THU 19 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
AF INEQUALITY IN Hn
In 2014, Guofang Wang and Chao Xia used the quermassintegralpreserving curvature flows to prove AF inequality for curvature integrals inHn:
THEOREM, WANG-XIA, 2014, ADV. MATH.
Let 1 ≤ k ≤ n − 1. If Ω ⊂ Hn is a bounded domain with Σ = ∂Ω h-convex,then ∫
Σ
pk ≥ ωn−1
[(|Σ|ωn−1
) 2k
+
(|Σ|ωn−1
) 2k
n−1−kn−1
] k2
,
Equality holds iff Σ is a geodesic sphere.
Remark. The Conjecture for 1 ≤ k ≤ n − 1 holds for h-convexhypersurfaces in Hn.
HMCF Haizhong Li THU 19 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
MAIN RESULTS
Now we present our main results.
The first result is
THEOREM A, ANDREWS-HU-L., ARXIV:1903.05903
Let 0 < 2k < n − 1. If Σ is a strictly convex hypersurface in Hn, then
∫Σ
pn−1−2k
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
k
.
Equality holds iff Σ is a geodesic sphere.
HMCF Haizhong Li THU 20 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
MAIN RESULTS
Now we present our main results.
The first result is
THEOREM A, ANDREWS-HU-L., ARXIV:1903.05903
Let 0 < 2k < n − 1. If Σ is a strictly convex hypersurface in Hn, then
∫Σ
pn−1−2k
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
k
.
Equality holds iff Σ is a geodesic sphere.
HMCF Haizhong Li THU 20 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
MAIN RESULTS
With the help of this AF inequality, we apply the inverse mean curvatureflow to prove
THEOREM B, ANDREWS-HU-L., ARXIV:1903.05903
Let n − 1 > 2. If Σ is a strictly convex hypersurface in Hn, then
∫Σ
pn−1 ≥ |Σ|
[1 +
(|Σ|ωn−1
)− 2n−1] n−1
2
.
Equality holds iff Σ is a geodesic sphere.
Remark. This verifies the Conjecture mentioned above for the casek = n − 1.
HMCF Haizhong Li THU 21 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
MAIN RESULTS
With the help of this AF inequality, we apply the inverse mean curvatureflow to prove
THEOREM B, ANDREWS-HU-L., ARXIV:1903.05903
Let n − 1 > 2. If Σ is a strictly convex hypersurface in Hn, then
∫Σ
pn−1 ≥ |Σ|
[1 +
(|Σ|ωn−1
)− 2n−1] n−1
2
.
Equality holds iff Σ is a geodesic sphere.
Remark. This verifies the Conjecture mentioned above for the casek = n − 1.
HMCF Haizhong Li THU 21 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
HARMONIC MEAN CURVATURE FLOW
Here we give the proofs of Theorems A & B.
The proof is the classical method for proving geometric inequalities byestablishing the monotonicity properties along a suitable curvature flow, andthe smooth convergence of this flow.
The key difference from the previous work is that we choose the contractingcurvature flow in Hn.
Given a smooth hypersurface Σ0 in Hn, parametrized by an embeddingX0 : Mn−1 → Hn. The harmonic mean curvature flow (HMCF) is a family ofembeddings X : Mn−1 × [0,T )→ Hn satisfying
∂
∂tX (x , t) =− pn−1
pn−2(x , t)ν(x , t),
X (·,0) =X0(·).
HMCF Haizhong Li THU 22 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
HARMONIC MEAN CURVATURE FLOW
Here we give the proofs of Theorems A & B.
The proof is the classical method for proving geometric inequalities byestablishing the monotonicity properties along a suitable curvature flow, andthe smooth convergence of this flow.
The key difference from the previous work is that we choose the contractingcurvature flow in Hn.
Given a smooth hypersurface Σ0 in Hn, parametrized by an embeddingX0 : Mn−1 → Hn. The harmonic mean curvature flow (HMCF) is a family ofembeddings X : Mn−1 × [0,T )→ Hn satisfying
∂
∂tX (x , t) =− pn−1
pn−2(x , t)ν(x , t),
X (·,0) =X0(·).
HMCF Haizhong Li THU 22 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
HARMONIC MEAN CURVATURE FLOW
Here we give the proofs of Theorems A & B.
The proof is the classical method for proving geometric inequalities byestablishing the monotonicity properties along a suitable curvature flow, andthe smooth convergence of this flow.
The key difference from the previous work is that we choose the contractingcurvature flow in Hn.
Given a smooth hypersurface Σ0 in Hn, parametrized by an embeddingX0 : Mn−1 → Hn. The harmonic mean curvature flow (HMCF) is a family ofembeddings X : Mn−1 × [0,T )→ Hn satisfying
∂
∂tX (x , t) =− pn−1
pn−2(x , t)ν(x , t),
X (·,0) =X0(·).
HMCF Haizhong Li THU 22 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
HARMONIC MEAN CURVATURE FLOW
Here we give the proofs of Theorems A & B.
The proof is the classical method for proving geometric inequalities byestablishing the monotonicity properties along a suitable curvature flow, andthe smooth convergence of this flow.
The key difference from the previous work is that we choose the contractingcurvature flow in Hn.
Given a smooth hypersurface Σ0 in Hn, parametrized by an embeddingX0 : Mn−1 → Hn. The harmonic mean curvature flow (HMCF) is a family ofembeddings X : Mn−1 × [0,T )→ Hn satisfying
∂
∂tX (x , t) =− pn−1
pn−2(x , t)ν(x , t),
X (·,0) =X0(·).
HMCF Haizhong Li THU 22 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
HARMONIC MEAN CURVATURE FLOW
Ben Andrews first proved the smooth convergence results for the flowof h-convex hypersurfaces in hyperbolic space, with speed given byfunctions with argument κi − 1, in particular the (shifted) harmonicmean curvature flow;
Later, Guoyi Xu proved the smooth convergence of the HMCF forstrictly convex hypersurfaces in hyperbolic space;Recently, Hao Yu proved the smooth convergence for a general classof contracting curvature flows in hyperbolic space.
A major ingredient in the proof of the smooth convergence of the HMCF isthe pinching estimate. That is, if the initial hypersurface Σ is strictly convex,then along the HMCF the evolving hypersurface Σt satisfies
κn−1(x , t) ≤ Cκ1(x , t), ∀(x , t) ∈ M × [0,T ∗),
where κ1 ≤ · · · ≤ κn−1 and C depends only on Σ.
HMCF Haizhong Li THU 23 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
HARMONIC MEAN CURVATURE FLOW
Ben Andrews first proved the smooth convergence results for the flowof h-convex hypersurfaces in hyperbolic space, with speed given byfunctions with argument κi − 1, in particular the (shifted) harmonicmean curvature flow;Later, Guoyi Xu proved the smooth convergence of the HMCF forstrictly convex hypersurfaces in hyperbolic space;
Recently, Hao Yu proved the smooth convergence for a general classof contracting curvature flows in hyperbolic space.
A major ingredient in the proof of the smooth convergence of the HMCF isthe pinching estimate. That is, if the initial hypersurface Σ is strictly convex,then along the HMCF the evolving hypersurface Σt satisfies
κn−1(x , t) ≤ Cκ1(x , t), ∀(x , t) ∈ M × [0,T ∗),
where κ1 ≤ · · · ≤ κn−1 and C depends only on Σ.
HMCF Haizhong Li THU 23 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
HARMONIC MEAN CURVATURE FLOW
Ben Andrews first proved the smooth convergence results for the flowof h-convex hypersurfaces in hyperbolic space, with speed given byfunctions with argument κi − 1, in particular the (shifted) harmonicmean curvature flow;Later, Guoyi Xu proved the smooth convergence of the HMCF forstrictly convex hypersurfaces in hyperbolic space;Recently, Hao Yu proved the smooth convergence for a general classof contracting curvature flows in hyperbolic space.
A major ingredient in the proof of the smooth convergence of the HMCF isthe pinching estimate. That is, if the initial hypersurface Σ is strictly convex,then along the HMCF the evolving hypersurface Σt satisfies
κn−1(x , t) ≤ Cκ1(x , t), ∀(x , t) ∈ M × [0,T ∗),
where κ1 ≤ · · · ≤ κn−1 and C depends only on Σ.
HMCF Haizhong Li THU 23 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
HARMONIC MEAN CURVATURE FLOW
Ben Andrews first proved the smooth convergence results for the flowof h-convex hypersurfaces in hyperbolic space, with speed given byfunctions with argument κi − 1, in particular the (shifted) harmonicmean curvature flow;Later, Guoyi Xu proved the smooth convergence of the HMCF forstrictly convex hypersurfaces in hyperbolic space;Recently, Hao Yu proved the smooth convergence for a general classof contracting curvature flows in hyperbolic space.
A major ingredient in the proof of the smooth convergence of the HMCF isthe pinching estimate. That is, if the initial hypersurface Σ is strictly convex,then along the HMCF the evolving hypersurface Σt satisfies
κn−1(x , t) ≤ Cκ1(x , t), ∀(x , t) ∈ M × [0,T ∗),
where κ1 ≤ · · · ≤ κn−1 and C depends only on Σ.
HMCF Haizhong Li THU 23 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
HARMONIC MEAN CURVATURE FLOW
Recall that the inner radius ρ− and outer radius ρ+ of a bounded domain Ωtwith boundary Σt in Hn is defined by
ρ−(t) := sup ρ : Bρ(p) is enclosed by Ωt for some p ∈ Hn ,ρ+(t) := inf ρ : Bρ(p) encloses Ωt for some p ∈ Hn ,
By the contracting property of the HMCF, together with the pinchingestimate, we prove that the inner radius and outer radius is comparable as itshrinks to a point.
LEMMA, ANDREWS-HU-L., ARXIV:1903.05903
Let Σt be a solution of the HMCF on a maximal time interval [0,T ∗). Thereexist positive constants C and η, depending only on the initial hypersurfaceΣ, such that
ρ+(t) ≤ Cρ−(t), ∀t ∈ [T ∗ − η,T ∗).
HMCF Haizhong Li THU 24 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
HARMONIC MEAN CURVATURE FLOW
Recall that the inner radius ρ− and outer radius ρ+ of a bounded domain Ωtwith boundary Σt in Hn is defined by
ρ−(t) := sup ρ : Bρ(p) is enclosed by Ωt for some p ∈ Hn ,ρ+(t) := inf ρ : Bρ(p) encloses Ωt for some p ∈ Hn ,
By the contracting property of the HMCF, together with the pinchingestimate, we prove that the inner radius and outer radius is comparable as itshrinks to a point.
LEMMA, ANDREWS-HU-L., ARXIV:1903.05903
Let Σt be a solution of the HMCF on a maximal time interval [0,T ∗). Thereexist positive constants C and η, depending only on the initial hypersurfaceΣ, such that
ρ+(t) ≤ Cρ−(t), ∀t ∈ [T ∗ − η,T ∗).
HMCF Haizhong Li THU 24 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
HARMONIC MEAN CURVATURE FLOW
Recall that the inner radius ρ− and outer radius ρ+ of a bounded domain Ωtwith boundary Σt in Hn is defined by
ρ−(t) := sup ρ : Bρ(p) is enclosed by Ωt for some p ∈ Hn ,ρ+(t) := inf ρ : Bρ(p) encloses Ωt for some p ∈ Hn ,
By the contracting property of the HMCF, together with the pinchingestimate, we prove that the inner radius and outer radius is comparable as itshrinks to a point.
LEMMA, ANDREWS-HU-L., ARXIV:1903.05903
Let Σt be a solution of the HMCF on a maximal time interval [0,T ∗). Thereexist positive constants C and η, depending only on the initial hypersurfaceΣ, such that
ρ+(t) ≤ Cρ−(t), ∀t ∈ [T ∗ − η,T ∗).
HMCF Haizhong Li THU 24 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
HARMONIC MEAN CURVATURE FLOW
Recall that the inner radius ρ− and outer radius ρ+ of a bounded domain Ωtwith boundary Σt in Hn is defined by
ρ−(t) := sup ρ : Bρ(p) is enclosed by Ωt for some p ∈ Hn ,ρ+(t) := inf ρ : Bρ(p) encloses Ωt for some p ∈ Hn ,
By the contracting property of the HMCF, together with the pinchingestimate, we prove that the inner radius and outer radius is comparable as itshrinks to a point.
LEMMA, ANDREWS-HU-L., ARXIV:1903.05903
Let Σt be a solution of the HMCF on a maximal time interval [0,T ∗). Thereexist positive constants C and η, depending only on the initial hypersurfaceΣ, such that
ρ+(t) ≤ Cρ−(t), ∀t ∈ [T ∗ − η,T ∗).
HMCF Haizhong Li THU 24 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
QUERMASSINTEGRALS IN Hn
For a convex domain Ω ⊂ Hn, the quermassintegrals are defined by
Wr (Ω) :=(n − r)ωr−1 · · ·ω0
nωn−2 · · ·ωn−r−1
∫Lχ(L ∩ Ω)dL, r = 1, · · · ,n − 1,
where Lr is the space of r -dim totally geodesic subspaces L in Hn, and dLis the natural measure on Lr which is invariant under the isometry group ofHn. The function χ is defined to be 1 if L ∩ Ω 6= ∅ and to be 0 otherwise.Furthermore, we set W0(Ω) = |Ω| and Wn(Ω) = ωn−1/n.
In Rn, the quermassintegrals coincide with the curvature integrals up to aconstant multiple. However, the quermassintegrals and the curvatureintegrals in Hn do not coincide. They are closely related by∫
Σ
pk = n(
Wk+1(Ω) +k
n − k + 1Wk−1(Ω)
), k = 1, · · · ,n − 1.
HMCF Haizhong Li THU 25 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
QUERMASSINTEGRALS IN Hn
For a convex domain Ω ⊂ Hn, the quermassintegrals are defined by
Wr (Ω) :=(n − r)ωr−1 · · ·ω0
nωn−2 · · ·ωn−r−1
∫Lχ(L ∩ Ω)dL, r = 1, · · · ,n − 1,
where Lr is the space of r -dim totally geodesic subspaces L in Hn, and dLis the natural measure on Lr which is invariant under the isometry group ofHn. The function χ is defined to be 1 if L ∩ Ω 6= ∅ and to be 0 otherwise.Furthermore, we set W0(Ω) = |Ω| and Wn(Ω) = ωn−1/n.
In Rn, the quermassintegrals coincide with the curvature integrals up to aconstant multiple. However, the quermassintegrals and the curvatureintegrals in Hn do not coincide. They are closely related by∫
Σ
pk = n(
Wk+1(Ω) +k
n − k + 1Wk−1(Ω)
), k = 1, · · · ,n − 1.
HMCF Haizhong Li THU 25 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – LIMITING BEHAVIOR
Since Wk is monotone under the set inclusion, i.e.,
Wk (Ω1) ≤Wk (Ω2), if Ω1 ⊂ Ω2,
we prove the following asymptotic behavior along the HMCF.
LEMMA, ANDREWS-HU-L., ARXIV:1903.05903
Let Σ be a strictly convex hypersurface in Hn. Let Σt , t ∈ [0,T ∗) be thesolution of the HMCF with the initial hypersurface Σ. Then we have
limt→T∗
∫Σt
pj =
0, 0 ≤ j ≤ n − 2;
ωn−1, j = n − 1,
HMCF Haizhong Li THU 26 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – LIMITING BEHAVIOR
Since Wk is monotone under the set inclusion, i.e.,
Wk (Ω1) ≤Wk (Ω2), if Ω1 ⊂ Ω2,
we prove the following asymptotic behavior along the HMCF.
LEMMA, ANDREWS-HU-L., ARXIV:1903.05903
Let Σ be a strictly convex hypersurface in Hn. Let Σt , t ∈ [0,T ∗) be thesolution of the HMCF with the initial hypersurface Σ. Then we have
limt→T∗
∫Σt
pj =
0, 0 ≤ j ≤ n − 2;
ωn−1, j = n − 1,
HMCF Haizhong Li THU 26 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
To prove Theorem A, we only need to find suitable monotone quantitiesalong the HMCF.
To prove the AF inequality
∫Σ
pn−1−2k
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
k
,
we consider the functional
Pk (t)
:=
(∫Σt
pn−1
ωn−1
)− n−1−2kn−1
∫
Σtpn−1−2k
ωn−1−
(∫Σt
pn−1
ωn−1
)1 −
(∫Σt
pn−1
ωn−1
)− 2n−1k .
HMCF Haizhong Li THU 27 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
To prove Theorem A, we only need to find suitable monotone quantitiesalong the HMCF.To prove the AF inequality
∫Σ
pn−1−2k
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
k
,
we consider the functional
Pk (t)
:=
(∫Σt
pn−1
ωn−1
)− n−1−2kn−1
∫
Σtpn−1−2k
ωn−1−
(∫Σt
pn−1
ωn−1
)1 −
(∫Σt
pn−1
ωn−1
)− 2n−1k .
HMCF Haizhong Li THU 27 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
To prove Theorem A, we only need to find suitable monotone quantitiesalong the HMCF.To prove the AF inequality
∫Σ
pn−1−2k
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
k
,
we consider the functional
Pk (t)
:=
(∫Σt
pn−1
ωn−1
)− n−1−2kn−1
∫
Σtpn−1−2k
ωn−1−
(∫Σt
pn−1
ωn−1
)1 −
(∫Σt
pn−1
ωn−1
)− 2n−1k .
HMCF Haizhong Li THU 27 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
Now we verify that Pk (t) is monotone increasing along the HMCF.
We first prove the case k = 1, i.e.,∫Σ
pn−3
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
.Along the HMCF, we have
ddt
∫Σ
pn−1 = −(n − 1)
∫Σ
pn−1,
and
ddt
∫Σ
pn−3 =− 2∫
Σ
pn−1 − (n − 3)
∫Σ
pn−1pn−4
pn−2
≥− 2∫
Σ
pn−1 − (n − 3)
∫Σ
pn−3.
HMCF Haizhong Li THU 28 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
Now we verify that Pk (t) is monotone increasing along the HMCF.
We first prove the case k = 1, i.e.,∫Σ
pn−3
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
.
Along the HMCF, we have
ddt
∫Σ
pn−1 = −(n − 1)
∫Σ
pn−1,
and
ddt
∫Σ
pn−3 =− 2∫
Σ
pn−1 − (n − 3)
∫Σ
pn−1pn−4
pn−2
≥− 2∫
Σ
pn−1 − (n − 3)
∫Σ
pn−3.
HMCF Haizhong Li THU 28 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
Now we verify that Pk (t) is monotone increasing along the HMCF.
We first prove the case k = 1, i.e.,∫Σ
pn−3
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
.Along the HMCF, we have
ddt
∫Σ
pn−1 = −(n − 1)
∫Σ
pn−1,
and
ddt
∫Σ
pn−3 =− 2∫
Σ
pn−1 − (n − 3)
∫Σ
pn−1pn−4
pn−2
≥− 2∫
Σ
pn−1 − (n − 3)
∫Σ
pn−3.
HMCF Haizhong Li THU 28 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
Then we have
ddt
(∫Σ
pn−1
ωn−1
)= −(n − 1)
(∫Σ
pn−1
ωn−1
),
and
ddt
∫Σpn−3
ωn−1−∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
≥− (n − 3)
∫Σpn−3
ωn−1−∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
.Therefore, we have
ddt
P1(t) ≥ 0.
By the limiting behavior of∫
Σtpj , we have limt→T∗ P1(t) = 0.
HMCF Haizhong Li THU 29 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
Then we have
ddt
(∫Σ
pn−1
ωn−1
)= −(n − 1)
(∫Σ
pn−1
ωn−1
),
and
ddt
∫Σpn−3
ωn−1−∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
≥− (n − 3)
∫Σpn−3
ωn−1−∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
.
Therefore, we haveddt
P1(t) ≥ 0.
By the limiting behavior of∫
Σtpj , we have limt→T∗ P1(t) = 0.
HMCF Haizhong Li THU 29 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
Then we have
ddt
(∫Σ
pn−1
ωn−1
)= −(n − 1)
(∫Σ
pn−1
ωn−1
),
and
ddt
∫Σpn−3
ωn−1−∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
≥− (n − 3)
∫Σpn−3
ωn−1−∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
.Therefore, we have
ddt
P1(t) ≥ 0.
By the limiting behavior of∫
Σtpj , we have limt→T∗ P1(t) = 0.
HMCF Haizhong Li THU 29 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
Thus we get P1(0) ≤ limt→T∗ P1(t) = 0, i.e.,
(∫Σ
pn−1
ωn−1
)− n−3n−1
∫Σpn−3
ωn−1−∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
≤ 0,
which is equivalent to∫Σ
pn−3
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
.
HMCF Haizhong Li THU 30 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
Thus we get P1(0) ≤ limt→T∗ P1(t) = 0, i.e.,
(∫Σ
pn−1
ωn−1
)− n−3n−1
∫Σpn−3
ωn−1−∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
≤ 0,
which is equivalent to∫Σ
pn−3
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
.
HMCF Haizhong Li THU 30 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
We prove the case k ≥ 2 by induction.
Assume that it holds for k − 1, i.e.,
(∫Σ
pn−1−2(k−1)
ωn−1
)≤(∫
Σpn−1
ωn−1
)1−(∫
Σpn−1
ωn−1
)− 2n−1
k−1
,
then we show that it also holds for k , i.e.,
∫Σ
pn−1−2k
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
k
.
HMCF Haizhong Li THU 31 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
We prove the case k ≥ 2 by induction.
Assume that it holds for k − 1, i.e.,
(∫Σ
pn−1−2(k−1)
ωn−1
)≤(∫
Σpn−1
ωn−1
)1−(∫
Σpn−1
ωn−1
)− 2n−1
k−1
,
then we show that it also holds for k , i.e.,
∫Σ
pn−1−2k
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
k
.
HMCF Haizhong Li THU 31 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
By the variational formula along the HMCF and Newton-MacLaurininequality, we have
ddt
∫Σ
pn−1−2k =− 2k∫
Σ
pn−2kpn−1
pn−2− (n − 1− 2k)
∫Σ
pn−2−2kpn−1
pn−2
≥− 2k∫
Σ
pn+1−2k − (n − 1− 2k)
∫Σ
pn−1−2k
=− 2k(∫
Σpn−1−2(k−1)
ωn−1
)− (n − 1− 2k)
(∫Σ
pn−1−2k
ωn−1
).
HMCF Haizhong Li THU 32 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
By the variational formula along the HMCF and Newton-MacLaurininequality, we have
ddt
∫Σ
pn−1−2k =− 2k∫
Σ
pn−2kpn−1
pn−2− (n − 1− 2k)
∫Σ
pn−2−2kpn−1
pn−2
≥− 2k∫
Σ
pn+1−2k − (n − 1− 2k)
∫Σ
pn−1−2k
=− 2k(∫
Σpn−1−2(k−1)
ωn−1
)− (n − 1− 2k)
(∫Σ
pn−1−2k
ωn−1
).
HMCF Haizhong Li THU 32 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
For simplicity, we take
x(t) =
∫Σ
pn−1
ωn−1, y(t) =
∫Σ
pn−1−2k
ωn−1.
Combining with induction on k − 1, we have
ddt
y =ddt
(∫Σ
pn−1−2k
ωn−1
)≥− 2k
(∫Σ
pn−1−2(k−1)
ωn−1
)− (n − 1− 2k)
(∫Σ
pn−1−2k
ωn−1
)≥− 2kx
(1− x−
2n−1
)k−1− (n − 1− 2k)y .
and ddt x = −(n − 1)x .
HMCF Haizhong Li THU 33 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
For simplicity, we take
x(t) =
∫Σ
pn−1
ωn−1, y(t) =
∫Σ
pn−1−2k
ωn−1.
Combining with induction on k − 1, we have
ddt
y =ddt
(∫Σ
pn−1−2k
ωn−1
)≥− 2k
(∫Σ
pn−1−2(k−1)
ωn−1
)− (n − 1− 2k)
(∫Σ
pn−1−2k
ωn−1
)≥− 2kx
(1− x−
2n−1
)k−1− (n − 1− 2k)y .
and ddt x = −(n − 1)x .
HMCF Haizhong Li THU 33 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
A direct calculation gives
ddt
[y − x
(1− x−
2n−1
)k]≥ −(n − 1− 2k)
[y − x
(1− x−
2n−1
)k],
and hence ddt Pk (t) ≥ 0.
By the limiting behavior of∫
Σtpj , we get
(∫Σ
pn−1
ωn−1
)− n−1−2kn−1
∫
Σpn−1−2k
ωn−1−(∫
Σpn−1
ωn−1
)1 −(∫
Σpn−1
ωn−1
)− 2n−1
k
=Pk (0) ≤ limt→T∗
Pk (t) = 0,
which is equivalent to
∫Σ
pn−1−2k
ωn−1≤(∫
Σpn−1
ωn−1
)1−(∫
Σpn−1
ωn−1
)− 2n−1
k
.
HMCF Haizhong Li THU 34 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
A direct calculation gives
ddt
[y − x
(1− x−
2n−1
)k]≥ −(n − 1− 2k)
[y − x
(1− x−
2n−1
)k],
and hence ddt Pk (t) ≥ 0. By the limiting behavior of
∫Σt
pj , we get
(∫Σ
pn−1
ωn−1
)− n−1−2kn−1
∫
Σpn−1−2k
ωn−1−(∫
Σpn−1
ωn−1
)1 −(∫
Σpn−1
ωn−1
)− 2n−1
k
=Pk (0) ≤ limt→T∗
Pk (t) = 0,
which is equivalent to
∫Σ
pn−1−2k
ωn−1≤(∫
Σpn−1
ωn−1
)1−(∫
Σpn−1
ωn−1
)− 2n−1
k
.
HMCF Haizhong Li THU 34 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM A – MONOTONICITY
A direct calculation gives
ddt
[y − x
(1− x−
2n−1
)k]≥ −(n − 1− 2k)
[y − x
(1− x−
2n−1
)k],
and hence ddt Pk (t) ≥ 0. By the limiting behavior of
∫Σt
pj , we get
(∫Σ
pn−1
ωn−1
)− n−1−2kn−1
∫
Σpn−1−2k
ωn−1−(∫
Σpn−1
ωn−1
)1 −(∫
Σpn−1
ωn−1
)− 2n−1
k
=Pk (0) ≤ limt→T∗
Pk (t) = 0,
which is equivalent to
∫Σ
pn−1−2k
ωn−1≤(∫
Σpn−1
ωn−1
)1−(∫
Σpn−1
ωn−1
)− 2n−1
k
.
HMCF Haizhong Li THU 34 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM B
Now we give the proof of Theorem B.
Here we use the inverse mean curvature flow (IMCF).∂
∂tX (x , t) =
1H(x , t)
ν(x , t),
X (·,0) =X0(·),
First, the strict convexity is preserved along the IMCF, so the followinginequality in Theorem A holds on the evolving hypersurfaces:∫
Σpn−3
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
.
HMCF Haizhong Li THU 35 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM B
Now we give the proof of Theorem B.Here we use the inverse mean curvature flow (IMCF).
∂
∂tX (x , t) =
1H(x , t)
ν(x , t),
X (·,0) =X0(·),
First, the strict convexity is preserved along the IMCF, so the followinginequality in Theorem A holds on the evolving hypersurfaces:∫
Σpn−3
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
.
HMCF Haizhong Li THU 35 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM B
Now we give the proof of Theorem B.Here we use the inverse mean curvature flow (IMCF).
∂
∂tX (x , t) =
1H(x , t)
ν(x , t),
X (·,0) =X0(·),
First, the strict convexity is preserved along the IMCF, so the followinginequality in Theorem A holds on the evolving hypersurfaces:∫
Σpn−3
ωn−1≤∫
Σpn−1
ωn−1
1−(∫
Σpn−1
ωn−1
)− 2n−1
.
HMCF Haizhong Li THU 35 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM B
By the variational formula along the IMCF and Newton-MacLaurininequality, we have
ddt
(|Σt |ωn−1
)=|Σt |ωn−1
,
ddt
(∫Σ
pn−1
ωn−1
)=
1ωn−1
∫Σ
pn−2
p1≤∫
Σpn−3
ωn−1
≤(∫
Σpn−1
ωn−1
)1−(∫
Σpn−1
ωn−1
)− 2n−1
.Hence, we consider the monotone increasing (we omit the proof) functional
Q(t) :=
(|Σt |ωn−1
)−1
|Σt |ωn−1
−
(∫Σt
pn−1
ωn−1
)1−
(∫Σt
pn−1
ωn−1
)− 2n−1
n−12 .
HMCF Haizhong Li THU 36 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM B
By the variational formula along the IMCF and Newton-MacLaurininequality, we have
ddt
(|Σt |ωn−1
)=|Σt |ωn−1
,
ddt
(∫Σ
pn−1
ωn−1
)=
1ωn−1
∫Σ
pn−2
p1≤∫
Σpn−3
ωn−1
≤(∫
Σpn−1
ωn−1
)1−(∫
Σpn−1
ωn−1
)− 2n−1
.Hence, we consider the monotone increasing (we omit the proof) functional
Q(t) :=
(|Σt |ωn−1
)−1
|Σt |ωn−1
−
(∫Σt
pn−1
ωn−1
)1−
(∫Σt
pn−1
ωn−1
)− 2n−1
n−12 .
HMCF Haizhong Li THU 36 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM B
By the variational formula along the IMCF and Newton-MacLaurininequality, we have
ddt
(|Σt |ωn−1
)=|Σt |ωn−1
,
ddt
(∫Σ
pn−1
ωn−1
)=
1ωn−1
∫Σ
pn−2
p1≤∫
Σpn−3
ωn−1
≤(∫
Σpn−1
ωn−1
)1−(∫
Σpn−1
ωn−1
)− 2n−1
.
Hence, we consider the monotone increasing (we omit the proof) functional
Q(t) :=
(|Σt |ωn−1
)−1
|Σt |ωn−1
−
(∫Σt
pn−1
ωn−1
)1−
(∫Σt
pn−1
ωn−1
)− 2n−1
n−12 .
HMCF Haizhong Li THU 36 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM B
By the variational formula along the IMCF and Newton-MacLaurininequality, we have
ddt
(|Σt |ωn−1
)=|Σt |ωn−1
,
ddt
(∫Σ
pn−1
ωn−1
)=
1ωn−1
∫Σ
pn−2
p1≤∫
Σpn−3
ωn−1
≤(∫
Σpn−1
ωn−1
)1−(∫
Σpn−1
ωn−1
)− 2n−1
.Hence, we consider the monotone increasing (we omit the proof) functional
Q(t) :=
(|Σt |ωn−1
)−1
|Σt |ωn−1
−
(∫Σt
pn−1
ωn−1
)1−
(∫Σt
pn−1
ωn−1
)− 2n−1
n−12 .
HMCF Haizhong Li THU 36 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM B
Now we analyze the asymptotics of Q(t) as t →∞.
We have |Σt | = |Σ|et . The convergence result of Gerhardt gives
hji =
(1 + O(e−
tn−1 )
)δj
i , on Σt .
As pn−1 is homogeneous of degree n − 1, we get
pn−1(hji ) = (1 + O(e−
tn−1 ))n−1 = 1 + O(e−
tn−1 ), on Σt .
and ∫Σt
pn−1
ωn−1=|Σt |ωn−1
(1 + O(e−
tn−1 )
)= O(et ), on Σt .
HMCF Haizhong Li THU 37 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM B
Now we analyze the asymptotics of Q(t) as t →∞.We have |Σt | = |Σ|et . The convergence result of Gerhardt gives
hji =
(1 + O(e−
tn−1 )
)δj
i , on Σt .
As pn−1 is homogeneous of degree n − 1, we get
pn−1(hji ) = (1 + O(e−
tn−1 ))n−1 = 1 + O(e−
tn−1 ), on Σt .
and ∫Σt
pn−1
ωn−1=|Σt |ωn−1
(1 + O(e−
tn−1 )
)= O(et ), on Σt .
HMCF Haizhong Li THU 37 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM B
Now we analyze the asymptotics of Q(t) as t →∞.We have |Σt | = |Σ|et . The convergence result of Gerhardt gives
hji =
(1 + O(e−
tn−1 )
)δj
i , on Σt .
As pn−1 is homogeneous of degree n − 1, we get
pn−1(hji ) = (1 + O(e−
tn−1 ))n−1 = 1 + O(e−
tn−1 ), on Σt .
and ∫Σt
pn−1
ωn−1=|Σt |ωn−1
(1 + O(e−
tn−1 )
)= O(et ), on Σt .
HMCF Haizhong Li THU 37 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM B
It follows that
Q(t) =1−
(∫Σt
pn−1
|Σt |
)1−
(∫Σt
pn−1
ωn−1
)− 2n−1
n−12
=1−(
1 + O(e−t
n−1 ))(
1 + O(e−2t
n−1 )) n−1
2
=1−(
1 + O(e−t
n−1 ))(
1 + O(e−2t
n−1 ))
=O(e−t
n−1 ),
which gives limt→∞Q(t) = 0.
Together with monotonicity of Q(t), we get
Q(0) ≤ Q(t) ≤ limt→∞
Q(t) = 0,
which is equivalent to∫Σ
pn−1 ≥ |Σ|
[1 +
(|Σ|ωn−1
)− 2n−1] n−1
2
.
HMCF Haizhong Li THU 38 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
PROOF OF THEOREM B
It follows that
Q(t) =1−
(∫Σt
pn−1
|Σt |
)1−
(∫Σt
pn−1
ωn−1
)− 2n−1
n−12
=1−(
1 + O(e−t
n−1 ))(
1 + O(e−2t
n−1 )) n−1
2
=1−(
1 + O(e−t
n−1 ))(
1 + O(e−2t
n−1 ))
=O(e−t
n−1 ),
which gives limt→∞Q(t) = 0. Together with monotonicity of Q(t), we get
Q(0) ≤ Q(t) ≤ limt→∞
Q(t) = 0,
which is equivalent to∫Σ
pn−1 ≥ |Σ|
[1 +
(|Σ|ωn−1
)− 2n−1] n−1
2
.
HMCF Haizhong Li THU 38 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
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Y. Ge, G. Wang and J. Wu, Hyperbolic Alexandorv-Fenchelquermassintegral inequality II, J. Differential Geom. 98(2014), 237-260.
C. Gerhardt, Inverse curvature flows in hyperbolic space, J. DifferentialGeom. 89(2011), 487-527.
HMCF Haizhong Li THU 39 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
Y. Hu and H. Li, Geometric inequalities for hypersurfaces withnonnegative sectional curvature in hyperbolic space, Calc. Var. PartialDiffernetial Equations 58(2019), no.2, 58:55.
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HMCF Haizhong Li THU 40 / 41
Iso. ineq. in Euclidean space AF ineq. in Euclidean space AF ineq. in hyperbolic space Main results
Thank you for your attention!
HMCF Haizhong Li THU 41 / 41