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Heming Jiao

Publications and source records attributed to Heming Jiao.

14 recordsLinked to original sources
arXiv2026

Interior Hessian estimates for Hessian quotient equations in dimension three

In this paper, we establish the interior Hessian estimates for $2$-convex solutions to $\frac{\sigma_2}{\sigma_1} (D^2 u) = \psi (x,u)$ in dimension three. In higher dimensions ($n \geq 4$), we prove the interior Hessian estimates for semi-convex solutions. We provide a new method to prove the doubling inequality for smooth solutions in dimensions three and four. In higher dimensions ($n\geq 5$) the doubling inequality is proved under an additional dynamic semi-convexity condition which is the same to that in \cite{SY2025}. The method also applies to the equation $\sigma_2 (D^2 u) = \psi (x, u, \nabla u)$.

math.AP↗
arXiv2026

Three Bernstein type theorems for hypersurfaces with zero Gaussian curvature

In this paper, we prove Bernstein type theorems for entire convex graphical hypersurfaces with zero Gaussian curvature in both Euclidean and Minkowski context. A supplementary example illustrates that zero Gaussian convex spacelike hypersurfaces are not necessary hyperplanes without additional conditions. We show that a zero Gaussian curvature convex hypersurface must be a hyperplane if the mean curvature goes to zero at infinity. In the Minkowski context, we prove similar results for hypersurface without timelike points.

math.DG↗
arXiv2025

Locally uniform ellipticity of the fractional Hessian operators

In [1], Caffarelli-Charro introduced a fractional Monge-Amp\`{e}re operator. Later, Wu [17] generalized it to a fractional analogue of $k$-Hessian operators and proved the strict ellipticity for $k=2$. In this paper, we introduce a fractional analogue of general Hessian operators and prove the stability. We also show that the fractional analogue $k$-Hessian operators defined in [17] are strictly elliptic with respect to convex solutions for all $2 \leq k \leq n$. Furthermore, we provide a new proof for the case $k=2$ without the convexity condition.

math.AP↗
arXiv2024

A geometric flow on noncompact affine Riemannian manifolds

In this paper, we obtain the existence criteria for a geometic flow on noncompact affine Riemannian manifolds. Our results can be regarded as a real version of Lee-Tam [19]. As an application, we prove that a complete noncompact Hessian manifold with nonnegative Hessian sectional curvature and bounded geometry is diffeomorphic to $\mathbb{R}^n$ if its tangent bundle has maximal volume growth.

math.DG↗
arXiv2022

The Pogorelov estimates for degenerate curvature equations

We establish the Pogorelov type estimates for degenerate prescribed k-curvature equations as well as k-Hessian equations. Furthermore,we investigate the interior C1,1 regularity of the solutions for Dirichlet problems. These techniques also enable us to improve the existence theorem for an asymptotic Plateau type problem in hyperbolic space.

math.AP↗
arXiv2022

Second order estimates for convex solutions of degenerate $k$-Hessian equations

The $C^{1,1}$ estimate of the Dirichlet problem for degenerate $k$-Hessian equations with non-homogenous boundary conditions is an open problem, if the right hand side function $f$ is only assumed to satisfy $f^{1/(k-1)} \in C^{1,1}$. In this paper, we solve this problem for convex solutions defined in the strictly convex bounded domain.

math.AP↗
arXiv2022

The Dirichlet problem for a class of prescribed curvature equations

In this paper, we consider the Dirichlet problem for a class of prescribed curvature equations. Both degenerate and non-degenerate cases are considered. The existence of the $C^{1,1}$ regular graphic hypersurfaces with prescribing a class of curvatures and constant boundary is proved for the degenerate case.

math.AP↗
arXiv2014

The Dirichlet problem for a class of Hessian type equations

We are concerned with the Dirichlet problem for a class of Hessian type equations. Applying some new methods we are able to establish the $C^2$ estimates for an approximating problem under essentially optimal structure conditions. Based on these estimates, the existence of classical solutions is proved.

math.AP↗
arXiv2014

Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds

We derive a priori estimates for second order derivatives of solutions to a wide calss of fully nonlinear elliptic equations on Riemannian manifolds. The equations we consider naturally appear in geometric problems and other applications such as optimal transportation. There are some fundamental assumptions in the literature to ensure the equations to be elliptic and that one can apply Evans-Krylov theorem once estimates up to second derivatives are derived. However, in previous work one needed extra assumptions which are more technical in nature to overcome various difficulties. In this paper we are able to remove most of these technical assumptions. Indeed, we derive the estimates under conditions which are almost optimal, and prove existence results for the Dirichlet problem which are new even for bounded domains in Euclidean space. Moreover, our methods can be applied to other types of nonlinear elliptic and parabolic equations, including those on complex manifolds.

math.AP↗