arXiv research

Study classifies 3D Hessian manifolds, proving their topology.

Study confirms Chern's conjecture on compact Hessian manifolds and classifies their topologies.

The Hesse-Koszul flow converges to the Hesse-Einstein metric on compact Hessian manifolds.

The note provides uniform estimates for complex Hessian equations on compact Hermitian manifolds.

Solves Monge-Ampère equations on compact Hessian manifolds using the Perron method.

Proves existence and uniqueness of viscosity solutions to complex Hessian equations on compact Hermitian manifolds.

The study finds continuous solutions to complex Hessian equations on compact Hermitian manifolds.

Paper studies Hessian quotient equations in warped product manifolds.

Study Hessian equations on compact Kähler manifolds with prescribed singularities.

We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge-Ampère, Hessian and inverse Hessian equations. As an application we solve a class of He…

Criterion for solvability of complex 2-Hessian equation on compact Kähler manifolds.

Kunneth formula derived for flat affine manifolds and applied to Hessian metrics.

Estimates for complex Hessian equations on Hermitian manifolds.

Let $(X,ω)$ be a compact Kähler manifold of dimension $n$ and fix $m\in \mathbb{N}$ such that $1\leq m \leq n$. We prove that any $(ω,m)$-sh function can be approximated from above by smooth $(ω,m)$-sh functions. A potential theory for the complex Hessian equation is also developed which generalizes the classical pluri…

Paper establishes estimates for solutions on compact manifolds.

Let $(X,ω)$ be a compact Kähler manifold of dimension $n$ and fix $1\leq m\leq n$. We prove that the total mass of the complex Hessian measure of $ω$-$m$-subharmonic functions is non-decreasing with respect to the singularity type. We then solve complex Hessian equations with prescribed singularity, and prove a Hodge i…

The paper defines Hesse solitons and explores their properties on Hessian manifolds.

Paper derives estimates for complex Hessian equations on Hermitian manifolds.

Paper proves inequalities on Hermitian manifolds with applications to bounded solutions.

Paper generalizes sub-slope definition and solves complex equations on compact manifolds.

We prove the existence of weak solutions of complex $m-$Hessian equations on compact Hermitian manifolds for the nonnegative right hand side belonging to $L^p, p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has been recently solved by Szekelyhidi and Zhang. We also give a stabilit…

Locally conformally Hessian manifolds are dense in radiant ones of rank 1.

Paper establishes estimates for nonlinear equations on compact manifolds.

We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian manifolds with boundary that equipped locally conformal Kähler metrics, provided a …

Study elliptic equations on hyperhermitian manifolds with flat hyperkähler metric.

The paper addresses geometric analysis on non-compact Riemannian manifolds, proving Calderón-Zygmund inequalities.

Study fully nonlinear elliptic equations on compact hyperhermitian manifolds.

Theory developed for complex Hessian measures on Hermitian manifolds.

Derives estimates for geometric elliptic equations on complex manifolds.

We prove a general inequality for mixed Hessian measures by global arguments. Our method also yields a simplification for the case of complex Monge-Ampère equation. Exploiting this and using Kołodziej's mass concentration technique we also prove the uniqueness of the solutions to the complex Hessian equation on compact…

The study finds starshaped compact hypersurfaces in warped products with curvature estimates.

New quasimetric spaces improve stability in complex Hessian equations.

We solve the Dirichlet problem for $k$-Hessian equations on compact complex manifolds with boundary, given the existence of a subsolution. Our method is based on a second order a priori estimate of the solution on the boundary with a particular gradient scale. The scale allows us to apply a blow-up argument to obtain c…

This paper has been withdrawn by the author due to a crucial error in the proof of Lemma 2.2.

In this paper we consider Monge-Ampère equations on compact Hessian manifolds, or equivalently Monge-Ampère equations on certain unbounded convex domains $Ω\subseteq \mathbb{R}^n$, with a periodicity constraint given by the action of an affine group. In the case where the affine group action is volume-preserving, i.e.,…

Let $(X,ω)$ be an $n$-dimensional compact Kähler manifold. We study degenerate complex Hessian equations of the form $(ω+dd^c\varphi)^m\wedge ω^{n-m}=F(x,\varphi)ω^n.$ Under some natural conditions on $F$, this equation has a unique continuous solution. When $(X,ω)$ is rational homogeneous we further show that the solu…

We study the Dirichlet problem of a class of fully nonlinear elliptic equations on Hermitian manifolds and derive a priori $C^2$ estimates which depend on the initial data on manifolds, the admissible subsolutions and the upper bound of the gradients of the solutions. In some special cases, we obtain the gradient estim…

Study on Hermitian Calabi functional in complexified orbits of symplectic manifolds.

In a complete Riemannian manifold $(M, g)$ if the hessian of a real valued function satisfies some suitable conditions then it restricts the geometry of $(M, g)$. In this paper we characterize all compact rank-1 symmetric spaces, as those Riemannian manifolds $(M, g)$ admitting a real valued function $u$ such that the …

Let $(X,ω)$ be a compact Kähler manifold of dimension $n$, and fix $1\leq m\leq n.$ We prove that the complex Hessian equation $(ω+dd^c\varphi)^m\wedge ω^{n-m}=fω^n$, with $0<f\in \mathcal{C}^{\infty}(X)$ has a smooth admissible solution $\varphi\in \mathcal{C}^{\infty}(X)$. This was previously known to hold when $(X,…

A C^2 function on C^n is called (n-1)-plurisubharmonic in the sense of Harvey-Lawson if the sum of any n-1 eigenvalues of its complex Hessian is nonnegative. We show that the associated Monge-Ampere equation can be solved on any compact Kahler manifold. As a consequence we prove the existence of solutions to an equatio…

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(…

The study proves that certain noncompact Hessian manifolds are diffeomorphic to R^n.

The study explores Hesse manifolds and their symmetries in multifield cosmological models.

Paper solves Hessian equations on Kähler manifolds.

A selfsimiar manifold is a Riemannian manifold $\left(M,g\right)$ endowed with a homothetic vector field $ξ$. We characterize global selfsimilar manifolds and describe the structure of local selfsimilar manifolds. We prove that any selfsimilar manifold with a potential homothetic vector field is a conical Riemannian ma…

New rigidity results for a generalized Ricci-Hessian equation on manifolds.

Study of contravariant pseudo-Hessian manifolds and their Poisson structures.