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Haizhong Li

Publications and source records attributed to Haizhong Li.

At least 55 records · Page 3Linked to original sources
arXiv2020

Locally constrained curvature flows and geometric inequalities in hyperbolic space

In this paper, we first study the locally constrained curvature flow of hypersurfaces in hyperbolic space, which was introduced by Brendle, Guan and Li [7]. This flow preserves the $m$th quermassintegral and decreases $(m+1)$th quermassintegral, so the convergence of the flow yields sharp Alexandrov-Fenchel type inequalities in hyperbolic space. Some special cases have been studied in [7]. In the first part of this paper, we show that h-convexity of the hypersurface is preserved along the flow and then the smooth convergence of the flow for h-convex hypersurfaces follows. We then apply this result to establish some new sharp geometric inequalities comparing the integral of $k$th Gauss-Bonnet curvature of a smooth h-convex hypersurface to its $m$th quermassintegral (for $0\leq m\leq 2k+1\leq n$), and comparing the weighted integral of $k$th mean curvature to its $m$th quermassintegral (for $0\leq m\leq k\leq n$). In particular, we give an affirmative answer to a conjecture proposed by Ge, Wang and Wu in 2015. In the second part of this paper, we introduce a new locally constrained curvature flow using the shifted principal curvatures. This is natural in the context of h-convexity. We prove the smooth convergence to a geodesic sphere of the flow for h-convex hypersurfaces, and provide a new proof of the geometric inequalities proved by Andrews, Chen and the third author of this paper in 2018. We also prove a family of new sharp inequalities involving the weighted integral of $k$th shifted mean curvature for h-convex hypersurfaces, which as application implies a higher order analogue of Brendle, Hung and Wang's [8] inequality.

math.DG↗
arXiv2020

Contraction of surfaces in hyperbolic space and in sphere

In this paper, we consider the contracting curvature flow of smooth closed surfaces in $3$-dimensional hyperbolic space and in $3$-dimensional sphere. In the hyperbolic case, we show that if the initial surface $M_0$ has positive scalar curvature, then along the flow by a positive power $α$ of the mean curvature $H$, the evolving surface $M_t$ has positive scalar curvature for $t>0$. By assuming $α\in [1,4]$, we can further prove that $M_t$ contracts a point in finite time and become spherical as the final time is approached. We also show the same conclusion for the flows by powers of scalar curvature and by powers of Gauss curvature provided that the power $α\in [1/2,1]$. In the sphere case, we show that the flow by a positive power $α$ of mean curvature contracts strictly convex surface in $\mathbb{S}^3$ to a round point in finite time if $α\in [1,5]$. The same conclusion also holds for the flow by powers of Gauss curvature provided that the power $α\in [1/2,1]$.

math.DG↗
arXiv2019

Minimal Lagrangian submanifolds of the complex hyperquadric

We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures. In particular, we define local angle functions encoding the geometry of the Lagrangian submanifold at hand. We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface. We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all, respectively all but one, local angle functions coincide.

math.DG↗
arXiv2019

Contracting axially symmetric hypersurfaces by powers of the $σ_k$-curvature

In this paper, we investigate the contracting curvature flow of closed, strictly convex axially symmetric hypersurfaces in $\mathbb{R}^{n+1}$ and $\mathbb{S}^{n+1}$ by $σ_k^α$, where $σ_k$ is the $k$-th elementary symmetric function of the principal curvatures and $α\ge 1/k$. We prove that for any $n\geq3$ and any fixed $k$ with $1\leq k\leq n$, there exists a constant $c(n,k)>1/k$ such that that if $α$ lies in the interval $[1/k,c(n,k)]$, then we have a nice curvature pinching estimate involving the ratio of the biggest principal curvature to the smallest principal curvature of the flow hypersurface, and we prove that the properly rescaled hypersurfaces converge exponentially to the unit sphere. In the case $1<k\le n \le k^2$, we can choose $c(n,k)=\frac{1}{k-1}$. Our results provide an evidence for the general convergence result without initial curvature pinching conditions.

math.DG↗
arXiv2019

Harmonic mean curvature flow and geometric inequalities

In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow to prove a new class of geometric inequalities for horospherically convex hypersurfaces in hyperbolic space. Using these new Alexandrov-Fenchel type inequalities and the inverse mean curvature flow, we obtain an Alexandrov-Fenchel inequality for strictly convex hypersurfaces in hyperbolic space, which was previously proved for horospherically convex hypersurfaces by Wang and Xia [44]. Finally, we use the mean curvature flow to prove a new Heintze-Karcher type inequality for hypersurfaces with positive Ricci curvature in hyperbolic space.

math.DG↗
arXiv2018

Surfaces expanding by non-concave curvature functions

In this paper, we first investigate the flow of convex surfaces in the space form $\mathbb{R}^3(κ)~(κ=0,1,-1)$ expanding by $F^{-α}$, where $F$ is a smooth, symmetric, increasing and homogeneous of degree one function of the principal curvatures of the surfaces and the power $α\in(0,1]$ for $κ=0,-1$ and $α=1$ for $κ=1$. By deriving that the pinching ratio of the flow surface $M_t$ is no greater than that of the initial surface $M_0$, we prove the long time existence and the convergence of the flow. No concavity assumption of $F$ is required. We also show that for the flow in $\mathbb{H}^3$ with $α\in (0,1)$, the limit shape may not be necessarily round after rescaling.

math.DG↗
arXiv2018

Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space

In this article, we will use inverse mean curvature flow to establish an optimal Sobolev-type inequality for hypersurfaces $Σ$ with nonnegative sectional curvature in $\mathbb{H}^n$. As an application, we prove the hyperbolic Alexandrov-Fenchel inequalities for hypersurfaces with nonnegative sectional curvature in $\mathbb{H}^n$: \begin{align*} \int_Σ p_{2k}\geq ω_{n-1}\left[\left(\frac{|Σ|}{ω_{n-1}}\right)^\frac{1}{k}+\left(\frac{|Σ|}{ω_{n-1}}\right)^{\frac{1}{k}\frac{n-1-2k}{n-1}}\right]^k, \end{align*} where $p_i$ is the normalized $i$-th mean curvature. Equality holds if and only if $Σ$ is a geodesic sphere in $\mathbb{H}^n$. For a domain $Ω\subset \mathbb{H}^n$ with $Σ=\partial Ω$ having nonnegative sectional curvature, we prove an optimal inequality for quermassintegral in $\mathbb{H}^n$: \begin{align*} W_{2k+1}(Ω)\geq \frac{ω_{n-1}}{n}\sum_{i=0}^{k}\frac{n-1-2k}{n-1-2i}C_k^i\left(\frac{|Σ|}{ω_{n-1}}\right)^\frac{n-1-2i}{n-1}, \end{align*} where $W_i(Ω)$ is the $i$-th quermassintegral in integral geometry. Equality holds if and only if $Σ$ is a geodesic sphere in $\mathbb{H}^n$. All these inequalities was previously proved by Ge, Wang and Wu \cite{Ge-Wang-Wu2014} under the stronger condition that $Σ$ is horospherical convex.

math.DG↗
arXiv2017

A Penrose type inequaltiy for graphs over Reissner-Nordström-anti-deSitter manifold

In this paper, we use the inverse mean curvature flow to establish an optimal Minkowski type inquality, weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Reissner-Nordström-anti-deSitter manifold and Penrose type inequality for asymptotically locally hyperbolic manifolds in which can be realized as graphs over Reissner-Nordström-anti-deSitter manifold.

math.DG↗
arXiv2017

Starshaped compact hypersurfaces with prescirbed Weingarten curvature in warped product manifolds

Given a compact Riemannian manifold $M$, we consider a warped product $\bar M = I \times_h M$ where $I$ is an open interval in $\Bbb R$. For a positive function $ψ$ defined on $\bar M$, we generalized the arguments in \cite{GRW2015} and \cite{RW16}, to obtain the curvature estimates for Hessian equations $σ_k(κ)=ψ(V,ν(V))$. We also obtain some existence results for the starshaped compact hypersurface $Σ$ satisfying the above equation with various assumptions.

math.AP↗
arXiv2017

Uniqueness of closed self-similar solutions to $σ_k^α$-curvature flow

By adapting the test functions introduced by Choi-Daskaspoulos \cite{c-d} and Brendle-Choi-Daskaspoulos \cite{b-c-d} and exploring properties of the $k$-th elementary symmetric functions $σ_{k}$ intensively, we show that for any fixed $k$ with $1\leq k\leq n-1$, any strictly convex closed hypersurface in $\mathbb{R}^{n+1}$ satisfying $σ_{k}^α=\langle X,ν\rangle$, with $α\geq \frac{1}{k}$, must be a round sphere. In fact, we prove a uniqueness result for any strictly convex closed hypersurface in $\mathbb{R}^{n+1}$ satisfying $F+C=\langle X,ν\rangle$, where $F$ is a positive homogeneous smooth symmetric function of the principal curvatures and $C$ is a constant.

math.DG↗
arXiv2017

A gap theorem for free boundary minimal surfaces in geodesic balls of hyperbolic space and hemisphere

In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the support function of the surface, and a natural potential function in hyperbolic space and hemisphere.

math.DG↗
arXiv2016

New characterizations of the Clifford torus as a Lagrangian self-shrinker

In this paper, we obtain several new characterizations of the Clifford torus as a Lagrangian self-shrinker. We first show that the Clifford torus $\mathbb{S}^1(1)\times\mathbb{S}^1(1)$ is the unique compact orientable Lagrangian self-shrinker in $\mathbb{C}^2$ with $|A|^2\leq 2$, which gives an affirmative answer to Castro-Lerma's conjecture. We also prove that the Clifford torus is the unique compact orientable embedded Lagrangian self-shrinker with nonnegative or nonpositive Gauss curvature in $\mathbb{C}^2$.

math.DG↗
arXiv2015

On inverse mean curvature flow in Schwarzschild space and Kottler space

In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface $Σ$ is strictly mean convex and star-shaped, then the flow hypersurface $Σ_t$ converges to a large coordinate sphere as $t\rightarrow \infty$ exponentially. We also describe an application of this convergence result. In the second part of this paper, we will analyse the inverse mean curvature flow in Kottler-Schwarzchild manifold. By deriving a lower bound for the mean curvature on the flow hypersurface independently of the initial mean curvature, we can use an approximation argument to show the global existence and regularity of the smooth inverse mean curvature flow for star-shaped and weakly mean convex initial hypersurface, which generalizes Huisken-Ilmanen's result [18].

math.DG↗
arXiv2014

Stability of capillary hypersurfaces in a Euclidean ball

We study the stability of capillary hypersurfaces in a unit Euclidean ball. It is proved that if the mass center of the generalized body enclosed by the immersed capillary hypersurface and the wetted part of the sphere is located at the origin, then the hypersurface is unstable. An immediate result is that all known examples except the totally geodesic ones and spherical caps are unstable.

math.DG↗
arXiv2013

Sharp diameter estimates for compact manifold with boundary

Let $(N,g)$ be an $n$-dimensional complete Riemannian manifold with nonempty boundary $\pt N$. Assume that the Ricci curvature of $N$ has a negative lower bound $Ric\geq -(n-1)c^2$ for some $c>0$, and the mean curvature of the boundary $\pt N$ satisfies $H\geq (n-1)c_0>(n-1)c$ for some $c_0>c>0$. Then a known result (see \cite{LN}) says that $\sup_{x\in N}d(x,\pt N)\leq \frac 1c\coth^{-1}\frac{c_0}c$. In this paper, we prove that if the boundary $\pt N$ is compact, then the equality holds if and only if $N$ is isometric to the geodesic ball of radius $\frac 1c\coth^{-1}\frac{c_0}c$ in an $n$-dimensional hyperbolic space $\mathbb{H}^n(-c^2)$ of constant sectional curvature $-c^2$. Moreover, we also prove an analogous result for manifold with nonempty boundary and with $m$-Bakry-Émery Ricci curvature bounded below by a negative constant.

math.DG↗