Convex Hypersurfaces in Hadamard Manifolds

On the asymptotic Plateau problem in Cartan--Hadamard manifolds

We prove dynamical stability of a natural class of hypersurface laminations defined over Cartan--Hadamard manifolds of pinched curvature. We achieve this by providing a complete solution to the asymptotic Plateau problem for immersed surfaces of constant extrinsic curvature in Cartan--Hadamard manifolds, proposed by Labourie in [25], together with its natural higher-dimensional generalisation.

Cartan-Hadamard流形中超曲面的凸性与刚性

Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds

本文证明Cartan-Hadamard流形中曲率在切平面消失的闭无穷小凸超曲面围成凸平坦区域，推广了经典刻画与刚性结果，并应用Gauss-Codazzi方程和Alexandrov几何技巧。

具有零化的Cartan-Hadamard流形中的等周和总曲率不等式

Isoperimetric and total curvature inequalities in Cartan-Hadamard manifolds with nullity

利用Chern-Gauss-Bonnet定理，在零化指标至少为n-3的Cartan-Hadamard流形M^n中，建立了凸超曲面的总Gauss-Kronecker曲率的尖锐不等式，从而将欧氏等周不等式推广到M^n，证明了这些空间的Cartan-Hadamard猜想。

等周不等式与Cartan-Hadamard流形中的常平均曲率超曲面

The isoperimetric inequality and CMC hypersurfaces in Cartan-Hadamard manifolds

本文通过常平均曲率超曲面的尖锐不等式及边界点对积分和Jacobi场估计，证明了$3\leq n\leq9$维Cartan-Hadamard流形中的欧几里得等周不等式，确立了相应维度的Cartan-Hadamard猜想。

旋转对称 Cartan-Hadamard 流形上的 Almgren 频率与 Agmon 凸性

Almgren frequency and Agmon convexity on rotationally symmetric Cartan-Hadamard manifolds

本文在旋转对称 Cartan-Hadamard 流形上引入全局频率函数，并在实双曲空间中证明相关频率函数的单调性，进而得到对数凸性定理和双曲三球定理。

When strictly locally convex hypersurfaces are embedded

In this paper we will prove Hadamard-Stoker type theorems in the following ambient spaces: $\man ^n \times \r$, where $\man ^n $ is a $1/4-$pinched manifold, and certain Killing submersions, e.g., Berger spheres and Heisenberg spaces. That is, under the condition that the principal curvatures of an immersed hypersurfaces are greater than some non-negative constant (depending on the ambient space), we prove that such a hypersurface is embedded and we also study its topology.

凸超曲面的刚性，常曲率多维空间中凸超曲面的刚性

Rigidity of convex hypersurfaces, Rigidity of convex hypersurfaces in multidimensional spaces of constant curvature

本文移除了Senkin推广Pogorelov刚性定理时的超曲面光滑性假设，建立了高维欧氏空间及其他常曲率模型空间中任意紧凸超曲面的内蕴度量刚性结果。

Locally convex hypersurfaces immersed in $H^n \times R$

We prove a theorem of Hadamard-Stoker type: a connected locally convex complete hypersurface immersed in $H^n \times R$ (n>1), where $H^n$ is n-dimensional hyperbolic space, is embedded and homeomorphic either to the n-sphere or to $R^n$. In the latter case it is either a vertical graph over a convex domain in $H^n$ or has what we call a simple end.

Curvature flow of complete convex hypersurfaces in hyperbolic space

We investigate the existence, convergence and uniqueness of modified general curvature flow of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.

A Survey on Convex Hypersurfaces of Riemannian Manifolds

We survey the main extensions of the classical Hadamard, Liebmann and Cohn-Vossen rigidity theorems on convex surfaces of $3$-Euclidean space to the context of convex hypersurfaces of Riemannian manifolds. The results we present include the one by Professor Renato Tribuzy (in collaboration with H. Rosenberg) on rigidity of convex surfaces of homogeneous $3$-manifolds.

常曲率空间中凸超曲面的局部刚性

Local rigidity of convex hypersurfaces in spaces of constant curvature

本文证明了常曲率空间（$n\ge4$）中凸等距超曲面在严格凸点附近局部全等，推广了Senkin在$C^1$光滑假设下的结果。

边界位于双曲面上的常曲率凸类空超曲面

Convex spacelike hypersurface of constant curvature with boundary on a hyperboloid

本文研究闵可夫斯基空间中边界在双曲面或光锥上的凸类空超曲面，证明在常高阶平均曲率且边界法向量夹角为常数的条件下，该超曲面必为另一双曲面的一部分。

On the Mean Curvature Flow for $σ_k$-Convex Hypersurfaces

We obtain estimates on both size and dimensions of the singular set at the first blow-up time of the mean curvature flow of hypersurfaces whose initial data is $σ_k$-convex.

Convex Hypersurfaces with Prescribed Scalar Curvature and Asymptotic Boundary in Hyperbolic Space

The existence of a smooth complete strictly locally convex hypersurface with prescribed scalar curvature and asymptotic boundary at infinity in $\mathbb{H}^{3}$ is proved under the assumption that there exists a strictly locally convex subsolution.

A note on minimal graphs over certain unbounded domains of Hadamard manifolds

Given an unbounded domain $Ω$ of a Hadamard manifold $M$, it makes sense to consider the problem of finding minimal graphs with prescribed continuous data on its cone-topology-boundary, i.e., on its ordinary boundary together with its asymptotic boundary. In this article it is proved that under the hypothesis that the sectional curvature of $M$ is $\le -1$ this Dirichlet problem is solvable if $Ω$ satisfies certain convexity condition at infinity and if $\partial Ω$ is mean convex. We also prove that mean convexity of $\partial Ω$ is a necessary condition, extending to unbounded domains some r…

Deforming a convex hypersurface with low entropy by its Gauss curvature

We prove the asymptotic roundness under normalized Gauss curvature flow provided entropy is initially small enough.

Classification of the locally strongly convex centroaffine hypersurfaces with parallel cubic form

In this paper, we study locally strongly convex centroaffine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the centroaffine metric. As the main result, we obtain a complete classification of such centroaffine hypersurfaces. The result of this paper is a centroaffine version of the complete classification of locally strongly convex equiaffine hypersurfaces with parallel cubic form due to Hu, Li and Vrancken [12].

Stability properties and topology at infinity of f-minimal hypersurfaces

We study stability properties of $f$-minimal hypersurfaces isometrically immersed in weighted manifolds with non-negative Bakry-Emery Ricci curvature under volume growth conditions. Moreover, exploiting a weighted version of a finiteness result and the adaptation to this setting of Li-Tam theory, we investigate the topology at infinity of $f$-minimal hypersurfaces. On the way, we prove a new comparison result in weighted geometry and we provide a general weighted $L^1$-Sobolev inequality for hypersurfaces in Cartan-Hadamard weighted manifolds, satisfying suitable restrictions on the weight fun…

Volume preserving non homogeneous mean curvature flow of convex hypersurfaces

We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed convex hypersurface converges to a round sphere. The proof is based on the monotonicity of the isoperimetric ratio, which allows to control the inner and outer radius of the hypersurface and to deduce uniform bounds on the curvature by maximum principle arguments.

The Non-Liner Dirichlet Problem in Hadamard Manifolds

We proof existence theorems for the Dirichlet problem for hypersurfaces of constant special Lagrangian curvature in Hadamard manifolds. The first results are obtained using the continuity method and approximation and then refined using two iterations of the Perron method. The a-priori estimates used in the continuity method are valid in any ambient manifold.

An optimal inequality on locally strongly convex centroaffine hypersurfaces

In this paper, we establish a general inequality for locally strongly convex centroaffine hypersurfaces in $\mathbb{R}^{n+1}$ involving the norm of the covariant derivatives of both the difference tensor $K$ and the Tchebychev vector field $T$. Our result is optimal in that, applying our recent classification for locally strongly convex centroaffine hypersurfaces with parallel cubic form in [4], we can completely classify the hypersurfaces which realize the equality case of the inequality.

具有挤压负曲率的Cartan-Hadamard流形中的总曲率与等周不等式

Total curvature and isoperimetric inequalities in pinched Cartan-Hadamard manifolds

本文针对具有挤压负曲率的Cartan-Hadamard流形，建立凸超曲面总Gauss-Kronecker曲率的尖锐下界，结合Chern-Gauss-Bonnet定理等完成证明，还应用该结果得到5维中Cartan-Hadamard猜想的等周不等式。

$H^α$-flow of mean convex, complete graphical hypersurfaces

We consider the evolution of hypersurfaces in $\mathbb{R}^{n+1}$ with normal velocity given by a positive power of the mean curvature. The hypersurfaces under consideration are assumed to be strictly mean convex (positive mean curvature), complete, and given as the graph of a function. Long-time existence of the $H^α$-flow is established by means of approximation by bounded problems.

常曲率凸超曲面奇性的传播

On the propagation of singularities of constant curvature, convex hypersurfaces

本文研究了双曲空间中常曲率凸超曲面的奇异集结构，并将其应用于解决一般凸曲率函数下的理想Plateau问题。

Some existence results to the Dirichlet problem for the minimal hypersurface equation on non mean convex domains of a Riemannian manifold

We prove the existence of minimal hypersurfaces for the Dirichlet that extends a similar result of Jenkins and Serrin in Euclidean Space to Riemannian ambient manifolds

Self-Similar Solutions to Curvature Flow of Convex Hypersurfaces

We classify the self-similar solutions to a class of Weingarten curvature flow of connected compact convex hypersurfaces, isometrically immersed into space forms with non-positive curvature, and obtain a new characterization of a sphere in a Euclidean space $\R^{n+1}$.

On an extention of the H^k mean curvature flow of closed convex hypersurfaces

In this paper we prove that the H^k (k is odd and larger than 2) mean curvature flow of a closed convex hypersurface can be extended over the maximal time provided that the total L^p integral of the mean curvature is finite for some p

A smooth foliation of convex hypersurfaces for quasi-hyper-Fuchsian manifolds

This submission has been withdrawn by the author and superseded by arXiv:0804.0744.

A geometric inequality for convex free boundary hypersurfaces in the unit ball

We use the inverse mean curvature flow with a free boundary perpendicular to the sphere to prove a geometric inequality involving the Willmore energy for convex hypersurfaces of dimension $n\geq 3$ with boundary on the sphere.

Starshaped locally convex hypersurfaces with prescribed curvature and boundary

In this paper we find strictly locally convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an application we show any smooth domain on the boundary of a compact strictly convex body can be deformed to a smooth hypersurface with the same boundary (inside the convex body) and realizing any prescribed curvature function smaller than the curvature of the body.

Minkowski type problems for convex hypersurfaces in the sphere

We consider the problem $F=f(ν)$ for strictly convex, closed hypersurfaces in $S^{n+1}$ and solve it for curvature functions $F$ the inverses of which are of class $(K)$.

Colding-Minicozzi Entropies in Cartan-Hadamard Manifolds

We introduce a family of functionals on submanifolds of Cartan-Hadamard manifolds that generalize the Colding-Minicozzi entropy of submanifolds of Euclidean space. We show that these functionals are monotone under mean curvature flow under natural conditions. As a consequence, we obtain sharp lower bounds on these entropies for certain closed hypersurfaces and observe a novel rigidity phenomenon.

Uniqueness of convex ancient solutions to hypersurface flows

We show that every convex ancient solution of mean curvature flow with Type I curvature growth is either spherical, cylindrical, or planar. We then prove the corresponding statement for flows by a natural class of curvature functions which are convex or concave in the second fundamental form. Neither of these results assumes interior noncollapsing.

Total mean curvatures of Riemannian hypersurfaces

We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces, via Reilly's identities. As applications we derive several geometric inequalities for a convex hypersurface $Γ$ in a Cartan-Hadamard manifold $M$. In particular we show that the first mean curvature integral of a convex hypersurface $γ$ nested inside $Γ$ cannot exceed that of $Γ$, which leads to a sharp lower bound in dimension $3$ for the total first mean curvature of $Γ$ in terms of the volume it bounds in $M$. This monotonicity property is extended to all mean curvature integrals when $γ$ is paralle…

凸超曲面上测地流的全局截面超曲面

Global Hypersurfaces of Section for Geodesic Flows on Convex Hypersurfaces

该研究针对容许等距对合的欧氏空间凸超曲面的测地流，构造出全局截面超曲面，实现了Birkhoff环面向高维的推广。

Convexity of $λ$-hypersurfaces

We prove that any $n$-dimensional closed mean convex $λ$-hypersurface is convex if $λ\le 0.$ This generalizes Guang's work on $2$-dimensional strictly mean convex $λ$-hypersurfaces. As a corollary, we obtain a gap theorem for closed $λ$-hypersurfaces with $λ\le 0.$

Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature

We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous functions of the principal curvatures of degree greater than one. For smooth, strictly convex initial hypersurfaces with ratio of principal curvatures sufficiently close to one at each point, we prove th…

The Dirichlet problem for prescribed curvature equations of $p$-convex hypersurfaces

In this paper, we study the Dirichlet problem for $p$-convex hypersurfaces with prescribed curvature. We prove that there exists a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition. An interior curvature estimate is also obtained.

Min-max minimal hypersurface in manifolds with convex boundary and $Ric\geq 0$

Let $(M^{n+1},\partial M,g)$ be a compact manifold with non-negative Ricci curvature, convex boundary and $2\leq n\leq 6$. We show that the min-max minimal hypersurface with respect to one-parameter families of hypersurfaces in $(M,\partial M)$ is orientable, of index one and multiplicity one.

Curvature estimates for $p$-convex hypersurfaces of prescribed curvature

In this paper, we establish the curvature estimates for $p$-convex hypersurfaces in $\mathbb{R}^{n+1}$ of prescribed curvature with $p\geq \frac{n}{2}$. The existence of a star-shaped hypersurface of prescribed curvature is obtained. We also prove a type of interior $C^2$ estimates for solutions to the Dirichlet problem of the corresponding equation.

Cyclic projections in Hadamard spaces

We show that cyclic products of projections onto convex subsets of Hadamard spaces can behave in a more complicated way than in Hilbert spaces, resolving a problem formulated by Miroslav Bačák. Namely, we construct an example of convex subsets in a Hadamard space such that the corresponding cyclic product of projections is not asymptotically regular.

Contracting convex hypersurfaces by curvature

We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smooth, and the limiting shape at the final time is spherical. We provide a surprisingly large family of flows for which such results fail, by a variety of mechanisms: Uniformly convex hypersurfaces may become non-convex, and smooth ones may develop curvature singularities; even where…

曲面面积与反向距离在凸超曲面中的研究

Area and antipodal distance in convex hypersurfaces

论文通过连续映射下的位移建立凸超曲面面积下界，并验证了Riemannian n球体积下界在所有维度中的有效性，同时给出了凸超曲面均宽的精确下界。

On the evolution of convex hypersurfaces by the $Q_k$ flow

We prove the existence and uniqueness of a $C^{1,1}$ solution of the $Q_k$ flow in the viscosity sense for compact convex hypersurfaces $Σ_t$ embedded in $R^{n+1}$ ($n \geq 2$) . In particular, for compact convex hypersurfaces with flat sides we show that, under a certain non-degeneracy initial condition, the interface separating the flat from the strictly convex side, becomes smooth, and it moves by the $Q_{k-1}$ flow at least for a short time.

整体非正曲率空间中Hermite-Hadamard型的分数阶积分估计

Fractional Integral Estimates of Hermite-Hadamard type in Global Nonpositive Curvature Spaces

该研究在整体非正曲率空间中引入测地$h$-凸函数以拓展函数凸性概念，通过Katugampola分数阶积分证明Hermite-Hadamard型估计，得到涉及平方距离映射的本质精确推论，丰富了弯曲几何空间的分析理论。

一类 Hessian 曲率方程的 k - 凸超曲面的存在性

The existence of $k$-convex hypersurface for a class of Hessian curvature equations

研究一类 Hessian 曲率方程闭星形超曲面存在性，结合先验估计与连续性方法建立 k - 凸超曲面存在唯一性，通过常秩定理证其严格凸。

Hypersurfaces with central convex cross-sections

A hypersurface $M$ in $\mathbb{R}^n$, $n \geq 4$, has central ovaloid property if $M$ intersects some hyperplane transversally along an ovaloid and every such ovaloid on $M$ has central symmetry. We show that a complete, connected, smooth hypersurface with central ovaloid property must either be a cylinder over a central ovaloid or else quadric.

Symmetric closed characteristics on symmetric compact convex hypersurfaces in $\mathbf{R}^8$: Special cases

Let $Σ$ be a $C^3$ compact symmetric convex hypersurface in $\mathbf{R}^{8}$. For some special cases, we prove that when $Σ$ carries exactly four geometrically distinct closed characteristics, then all of them must be symmetric.

Absence of bubbling phenomena for non convex anisotropic nearly umbilical and quasi Einstein hypersurfaces

We prove that, for every closed (not necessarily convex) hypersurface $Σ$ in $\mathbb{R}^{n+1}$ and every $p>n$, the $L^p$-norm of the trace-free part of the anisotropic second fundamental form controls from above the $W^{2,p}$-closeness of $Σ$ to the Wulff shape. In the isotropic setting, we provide a simpler proof. This result is sharp since in the subcritical regime $p\leq n$, the lack of convexity assumptions may lead in general to bubbling phenomena. Moreover, we obtain a stability theorem for quasi Einstein (not necessarily convex) hypersurfaces and we improve the quantitative estimates …

凸超曲面中的平均曲率与闭测地线

Mean curvature and closed geodesics in convex hypersurfaces

本文通过Birkhoff不变量给出了欧氏空间中凸超曲面的平均宽度下界，进而得到总平均曲率的尖锐下界，并刻画了球面为唯一使所有含最长弦的平面截面尽可能长的凸超曲面。

Deformations of unbounded convex bodies and hypersurfaces

We study the topology of the space $\d\K^n$ of complete convex hypersurfaces of $\R^n$ which are homeomorphic to $\R^{n-1}$. In particular, using Minkowski sums, we construct a deformation retraction of $\d\K^n$ onto the Grassmannian space of hyperplanes. So every hypersurface in $\d \K^n$ may be flattened in a canonical way. Further, the total curvature of each hypersurface evolves continuously and monotonically under this deformation. We also show that, modulo proper rotations, the subspaces of $\d\K^n$ consisting of smooth, strictly convex, or positively curved hypersurfaces are each contra…

Hadamard流形上的水平凸优化 第一部分：分析与算法

Horospherically Convex Optimization on Hadamard Manifolds Part I: Analysis and Algorithms

本文针对Hadamard流形上的优化问题，提出水平凸性（h-凸性）概念，并设计曲率无关的次梯度下降与Nesterov加速算法，匹配欧氏空间复杂度，并扩展到函数和最小化场景。