arXiv research

Study estimates eigenvalues for concave Hessian operators on convex domains.

Establishes a concavity property for positive Hessian quotient operators.

Estimates for complex Hessian equations on Hermitian manifolds.

Derives concavity inequality and estimates for $k$-Hessian equations.

We explain a general construction through which concave elliptic operators on complex manifolds give rise to concave functions on cohomology. In particular, this leads to generalized versions of the Khovanskii-Teissier inequalities.

Paper derives estimates for Hessian equations under concavity assumptions.

Paper estimates curvature of semi-convex solutions in hyperbolic space.

We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the $C^{1,1}$ boundary data, we establish the interior $C^{1,1}$-regularity of the unique (admissible) solution, which is o…

Study proves estimate for Hessian quotient equations on 2D Riemannian manifolds.

Proof that certain complex equations have only simple solutions.

Study concavity of solutions to elliptic equations under conformal deformations.

Establishes log-concavity estimates for convex domains' first Dirichlet eigenfunctions.

Established concavity principle for curved spaces.

This study solves a Dirichlet problem for specific elliptic equations on Riemannian manifolds with concave boundaries.

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C^{1+ε}, showing the opt…

We show that for any $δ\in [0,1)$ there exists a homogeneous order $2-δ$ analytic outside zero solution to a uniformly elliptic Hessian equation in R^5.

We use the octonion algebra to construct singular solutions of Hessian fully nonlinear uniformly elliptic equations in 21 or more dimensions. The regularity of these solutions is the least possible one. The same is proven for Isaacs equtions.

Derives estimates for geometric elliptic equations on complex manifolds.

Solves curvature problems on manifolds with negative curvature.

New method for analyzing elliptic and parabolic equations.

Solves open problems for fully nonlinear elliptic equations on manifolds.

Established in the 30's, Schauder {\it a priori} estimates are among the most classical and powerful tools in the analysis of problems ruled by 2nd order elliptic PDEs. Since then, a central problem in regularity theory has been to understand Schauder type estimates fashioning particular borderline scenarios. In such c…

Study elliptic equations on hyperhermitian manifolds with flat hyperkähler metric.

We show how to construct a non-smooth solution to Hessian fully nonlinear second-order uniformly elliptic equation using the Cartan isoparametric cubic in 5 dimensions.

In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some esti…

Let $u$ denote a solution to a rotationally invariant Hessian equation $F(D^2u)=0$ on a bounded simply connected domain $Ω\subset R^2$, with constant Dirichlet and Neumann data on $\partial Ω$. In this paper we prove that if $u$ is real analytic and not identically zero, then $u$ is radial and $Ω$ is a disk. The fully …

In this paper, we study Hessian equations and complex quotient equations on closed Hermitian manifolds. We directly derive the uniform estimate for the admissible solution. As an application, we solve general Hessian equations on closed Kähler manifolds.

Study fully nonlinear elliptic equations on compact hyperhermitian manifolds.

By adapting methods of \cite{AC} we prove a sharp estimate on the expansion modulus of the gradient of the log of the parabolic kernel to the Schördinger operator with convex potential, which improves an earlier work of Brascamp-Lieb. We also include alternate proofs to the improved log-concavity estimate, and to the f…

MALA mixes efficiently under smoothness and isoperimetry assumptions.

Let L be an ample bundle over a compact complex manifold X. Fix a Hermitian metric in L whose curvature defines a Kähler metric on X. The Hessian of Mabuchi energy is a fourth-order elliptic operator D on functions which arises in the study of scalar curvature. We quantise D by the Hessian E(k) of balancing energy, a f…

This research accelerates sampling methods using Nesterov's Acceleration.

We prove a priori interior $C^{2,α}$ estimates for solutions of fully nonlinear elliptic equations of twisted type. For example, our estimates apply to equations of the type convex + concave. These results are particularly well suited to equations arising from elliptic regularization. As application, we obtain a new pr…

We present a method to derive local estimates for some classes of fully nonlinear elliptic equations. The advantage of our method is that we derive Hessian estimates directly from $C^0$ estimates. Also, the method is flexible and can be applied to a large class of equations.

Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.

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…

A new sampling method, RC-LMC, reduces computational cost for high-dimensional log-concave distributions.

Optimistic method adapted for faster convex-concave min-max problems.

BBVI converges nearly dimensionally independent for log-concave targets.

Study on eigenvalues of complex Hessian operator on pseudoconvex manifolds.

This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using $KK$-theory.

In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure $γ$ enjoys $\frac{1}{n}$-concavity with respect to the Minkowski addition of \textbf{symmetric} convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex…

Paper discusses solving generalized Hessian inequalities with various operators.

Mathai, Melrose, and Singer introduced the notion of projective elliptic operators on manifolds equipped with an Azumaya bundle. In this note we compute the equivariant index of transversally elliptic operators that are the pullback of projective elliptic operators on the trivialization of the Azumaya bundle. It encomp…

Paper establishes estimates for solutions on compact manifolds.

New algorithms solve complex minimax problems efficiently.

We study entire continuous viscosity solutions to fully nonlinear elliptic equations involving the conformal Hessian. We prove the strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations when one of the competitors is $C^{1,1}$. We obtain as a consequence a Liouville theorem for entire solutio…

A new method solves a complex optimization problem efficiently.