arXiv research

New solutions found for elliptic systems with mixed couplings.

We prove that there is no nontrivial homogeneous order 2 solutions of fully nonlinear uniformly elliptic equations in dimension 4.

The study proves conditions for nontrivial solutions on Riemannian manifolds.

New domains found in hyperbolic space solve a specific elliptic problem.

We find exact solutions describing Ricci flows of four dimensional pp-waves nonlinearly deformed by two/three dimensional solitons. Such solutions are parametrized by five dimensional metrics with generic off-diagonal terms and connections with nontrivial torsion which can be related, for instance, to antisymmetric ten…

Neural networks solve the Dirichlet problem for Monge-Ampère equations.

New solutions found for Ginzburg-Landau equations on complex manifolds.

We are concerned with fully nonlinear elliptic equations on complex manifolds and search for technical tools to overcome difficulties in deriving a priori estimates which arise due to the nontrivial torsion and curvature, as well as the general (non-pseudoconvex) shape of the boundary. We present our methods, which wor…

We use min-max techniques to produce nontrivial solutions $u_ε:M\to \mathbb{R}^2$ of the Ginzburg-Landau equation $Δu_ε+\frac{1}{ε^2}(1-|u_ε|^2)u_ε=0$ on a given compact Riemannian manifold, whose energy grows like $|\logε|$ as $ε\to 0$. When the degree one cohomology $H^1_{dR}(M)=0$, we show that the energy of these s…

In this paper we explore the connection between special degenerations of algebraic manifolds and geodesics in the space of Kahler metrics. We provide a new and general geometric construction of nontrivial solutions for the geodesic equation. We show how to associate to any special nontrivial degeneration a geodesic of …

Study improves understanding of solutions to complex equations in geometry.

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

New action functionals for sigma models on Jacobi bundles.

We prove the existence of non-smooth solutions to fully nonlinear uniformly elliptic equations.

A metric projective structure is a manifold equipped with the unparametrised geodesics of some pseudo-Riemannian metric. We make acomprehensive treatment of such structures in the case that there is a projective Weyl curvature nullity condition. The analysis is simplified by a fundamental and canonical 2-tensor invaria…

The paper studies Vafa-Witten equations on Kaehler manifolds and identifies obstructions to nontrivial solutions.

This paper concerns local gradient estimates to solutions of general conformally invariant fully nonlinear elliptic equations of second order.

Classifies ancient solutions to curvature flows, finding two main types.

Using the `Riemann Problem with zeros' method, Ward has constructed exact solutions to a (2+1)-dimensional integrable Chiral Model, which exhibit solitons with nontrivial scattering. We give a correspondence between what we conjecture to be all pure soliton solutions and certain holomorphic vector bundles on a compact …

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.

We construct a new class of exact solutions describing spacetimes possessing Lie algebroid symmetry. They are described by generic off-diagaonal 5D metrics embedded in bosonic string gravity and possess nontrivial limits to the Einstein gravity. While we focus on nonholonomic vielbein transforms of the Schwarzschild me…

We observe that the comparison result of Barles-Biton-Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow.

We outline a new geometric method of constructing exact solutions of gravitational field equations parametrized by generic off-diagonal metrics, anholonomic frames and possessing, in general, nontrivial torsion and nonmetricity. The formalism of nonlinear connections is elaborated for (pseudo) Riemannian and Einstein-C…

Proves solutions to elliptic equations on Hermitian manifolds with optimal conditions.

A nontrivial smooth steady incompressible Euler flow in three dimensions with compact support is constructed. Another uncommon property of this solution is the dependence between the Bernoulli function and the pressure.

We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on the Euclidean space and on compact Riemannian manifolds. As a byproduct of the proofs we also obtain some rigidity, or partial symmetry…

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…

In this paper we provide a characterization of second order fully nonlinear CR invariant equations on the Heisenberg group, which is the analogue in the CR setting of the result proved in the Euclidean setting by A. Li and the first author (2003). We also prove a comparison principle for solutions of second order fully…

We study complete Riemannian manifolds satisfying the equation $Ric+\nabla^2 f-\frac{1}{m}df\otimes df=0$ by studying the associated PDE $Δ_f f + mμe^{2f/m}=0$ for $μ\leq 0$. By developing a gradient estimate for $f$, we show there are no nonconstant solutions. We then apply this to show that there are no nontrivial Ri…

Ancient pancake solutions found for curvature flows.

In this article we consider nonholonomic deformations of disk solutions in general relativity to generic off-diagonal metrics defining knew classes of exact solutions in 4D and 5D gravity. These solutions possess Lie algebroid symmetries and local anisotropy and define certain generalizations of manifolds with Killing …

Persistent elements are ubiquitous in knot groups, especially for hyperbolic knots.

The paper studies fully nonlinear equations on Hermitian manifolds, proving existence and interior estimates.

Solves a specific Dirichlet problem on Riemannian manifolds.

The paper proves estimates for solutions to nonlinear equations on manifolds with boundary.

We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.

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…

Study shows long-term solutions for complex equations on curved spaces.

New study confirms some mean curvature flow solutions have bounded mean curvature.

Paper studies solutions to a specific equation in conformal geometry with singular sets.

The paper establishes boundary estimates for solutions to elliptic equations on Hermitian manifolds.

In this paper, we consider the indefinite scalar curvature problem on $R^n$. We propose new conditions on the prescribing scalar curvature function such that the scalar curvature problem on $R^n$ (similarly, on $S^n$) has at least one solution. The key observation in our proof is that we use the bifurcation method to g…

Conformally equivariant quantization is a peculiar map between symbols of real weight $δ$ and differential operators acting on tensor densities, whose real weights are designed by $λ$ and $λ+δ$. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight $δ$. Later, Si…

We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positiv…

Paper proves no nontrivial solutions to certain elliptic equations on graphs.

Method discovers symmetries in data with neural networks.

Proposes a new framework for optimizing utility with state-dependent benchmarks.

Study proves radial symmetry of solutions to certain nonlinear equations in space forms.