arXiv subjects

Zhizhang Wang

Publications and source records attributed to Zhizhang Wang.

30 records · Page 2Linked to original sources

arXiv2018

Uniqueness of isometric immersions with the same mean curvature

Motivated by the quasi-local mass problem in general relativity, we study the rigidity of isometric immersions with the same mean curvature into a warped product space. As a corollary of our main result, two star-shaped hypersurfaces in a spatial Schwarzschild or AdS-Schwarzschild manifold with nonzero mass differ only by a rotation if they are isometric and have the same mean curvature. We also give similar results if the mean curvature condition is replaced by an $σ_2$-curvature condition.

math.DG↗

arXiv2017

The curvature estimates for convex solutions of some fully nonlinear Hessian type equations

The curvature estimates of quotient curvature equation do not always exist even for convex setting \cite{GRW}. Thus it is natural question to find other type of elliptic equations possessing curvature estimates. In this paper, we discuss the existence of curvature estimate for fully nonlinear elliptic equations defined by symmetric polynomials, mainlly, the linear combination of elementary symmetric polynomials.

math.AP↗

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↗

arXiv2016

The Weyl problem in warped product space

In this paper, we discuss the Weyl problem in warped product space. We obtain the openness, non rigidity and some applications. These results together with the a priori estimates obtained by Lu imply some existence results. Meanwhile we reprove the infinitesimal rigidity in the space forms.

math.DG↗

arXiv2016

On the curvature estimates for Hessian equations

The curvature estimates of $k$ curvature equations for general right hand side is a longstanding problem. In this paper, we totally solve the $n-1$ case and we also discuss some applications for our estimate.

math.AP↗

arXiv2015

An interior estimate for convex solutions and a rigidity theorem

We establish an interior $C^2$ estimate for $k+1$ convex solutions to Dirichlet problems of $k$-Hessian equations. We also use such estimate to obtain a rigidity theorem for $k+1$ convex entire solutions of $k$-Hessian equations in Euclidean space.

math.AP↗

arXiv2013

A proof of the Alexanderov's uniqueness theorem for convex surfaces in $\mathbb R^3$

We give a new proof of a classical uniqueness theorem of Alexandrov using the weak uniqueness continuation theorem of Bers-Nirenberg. We prove a version of this theorem with the minimal regularity assumption: the spherical hessians of the corresponding convex bodies as Radon measures are nonsingular.

math.AP↗

arXiv2013

Global $C^2$ estimates for convex solutions of curvature equations

We establish $C^2$ a priori estimate for convex hypersurfaces whose principal curvatures $κ=(κ_1,..., κ_n)$ satisfying Weingarten curvature equation $σ_k(κ(X))=f(X,ν(X))$. We also obtain such estimate for admissible 2-convex hypersurfaces in the case $k=2$. Our estimates resolve a longstanding problem in geometric fully nonlinear elliptic equations.

math.AP↗

arXiv2011

Form-type Calabi-Yau equations on Kähler manifolds of nonnegative orthogonal bisectional curvature

In this paper we prove the existence and uniqueness of the form-type Calabi-Yau equation on Kähler manifolds of nonnegative orthogonal bisectional curvature.

math.DG↗

arXiv2011

Semilinear equations, the $γ_k$ function, and generalized Gauduchon metrics

In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the $γ_k$ functions on the space of its hermitian metrics.

math.DG↗

arXiv2011

On the self-shrinking systems in arbitrary codimension spaces

In this paper, we discuss the self-shrinking systems in higher codimensional spaces. We mainly obtain several Bernstein type results and a sharp growth estimate.

math.DG↗

arXiv2010

Form-Type Calabi-Yau Equations

Motivated from mathematical aspects of the superstring theory, we introduce a new equation on a balanced, hermitian manifold, with zero first Chern class. Solving the equation, one will obtain, in each Bott--Chern cohomology class, a balanced metric which is hermitian Ricci--flat. This can be viewed as a differential form level generalization of the classical Calabi--Yau equation. We establish the existence and uniqueness of the equation on complex tori, and prove certain uniqueness and openness on a general Kähler manifold.

math.DG↗