arXiv research

The study provides interior curvature estimates for convex graphs satisfying a specific quotient equation.

Paper studies Hessian quotient equations in warped product manifolds.

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

Estimates solutions to Hessian quotient equations on HKT manifolds.

Paper solves Hessian quotient equations in Lorentz-Minkowski space with Dirichlet boundary conditions.

Paper solves curvature equations in Minkowski space for non-convex domains.

The paper proves a numerical condition for solving complex Hessian quotient equations with Calabi symmetry.

Interior $C^{2}$ estimates for sum Hessian quotient equations on Riemannian manifolds

We prove estimates for the sectional curvature of hyperkaehler quotients and give applications to moduli spaces of solutions to Nahm's equations and Hitchin's equations.

Solves second-order PDEs using quotients and differential invariants.

The paper establishes Pogorelov type estimates for solutions to Hessian quotient equations in hyperbolic space.

We present a novel formulation of the instanton equations in 8-dimensional Yang-Mills theory. This formulation reveals these equations as the last member of a series of gauge-theoretical equations associated with the real division algebras, including flatness in dimension 2 and (anti-)self-duality in 4. Using this form…

Establishes a concavity property for positive Hessian quotient operators.

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

In this paper, we consider the $V$-soliton equation which is a degenerate fully nonlinear equation introduced by La Nave and Tian in their work on Kähler-Ricci flow on symplectic quotients. One can apply the interpretation to study finite time singularities of the Kähler-Ricci flow. As in the case of Kähler-Einstein me…

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…

The paper solves the conjugacy problem in a specific braid group quotient and finds infinite virtually cyclic subgroups.

We show that there is no algorithm deciding whether the maximal residually free quotient of a given finitely presented group is finitely presentable or not. Given a finitely generated subgroup G of a finite product of limit groups, we discuss the possibility of finding an explicit set of defining equations (i.e. of exp…

We study Deraux's non arithmetic orbifold ball quotient surfaces obtained as birational transformations of a quotient $X$ of a particular Abelian surface $A$. Using the fact that $A$ is the Jacobian of the Bolza genus $2$ curve, we identify $X$ as the weighted projective plane $\mathbb{P}(1,3,8)$. We compute the equati…

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.

Paper solves Hessian equations on Kähler manifolds.

Positive mass theorem and Yamabe equation on CR manifolds

Using Traizet's regeneration method, we prove the existence of many new 3-dimensional families of embedded, doubly periodic minimal surfaces. All these families have a foliation of 3-dimensional Euclidean space by vertical planes as a limit. In the quotient, these limits can be realized conformally as noded Riemann sur…

Geometric analysis proves weak KAM solutions constant under specific conditions.

Develops finite-gap solutions for Pohlmeyer--Lund--Regge equation and Lund--Regge curve evolution.

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…

Solves critical LYZ equation in Kähler geometry.

We construct solutions to the constraint equations in general relativity using the limit equation criterion introduced by Dahl, Humbert and the first author. We focus on solutions over compact 3-manifolds admitting a $\bS^1$-symmetry group. When the quotient manifold has genus greater than 2, we obtain strong far from …

We present generating functions for extensions of multiplicative invariants of wreath symmetric products of orbifolds presented as the quotient by the locally free action of a compact, connected Lie group in terms of orbifold sector decompositions. Particularly interesting instances of these product formulas occur for …

The abstract describes a new manifold structure for NLS type equations.

We studied isometric stochastic flows of a Stratonovich stochastic differential equation on spheres, i.e. on the standard sphere and Gromoll-Meyer exotic sphere. The standard sphere $S^7_s$ can be constructed as the quotient manifold $\mathrm{Sp}(2, \mathbb{H})/S^3$ with the so-called ${\bullet}$-action of $S^3$, where…

The abstract shows how constant mean curvature surfaces in hyperbolic space are linked to the Liouville equation.

We classify solvable Lie groups admitting left invariant symplectic half-flat structure. When the Lie group has a compact quotient by a lattice, we show that these structures provide solutions of supersymmetric equations of type IIA.

It is proved that the only geodesically complete stationary vacuum solution of the Einstein equations is the empty Minkowski space, or a quotient of it by a discrete group of isometries, generalizing a classical result of Lichnerowicz. In addition, we obtain an apriori bound on the curvature of stationary vacuum soluti…

Solves open problems for fully nonlinear elliptic equations on manifolds.

Researchers create minimal surfaces with Scherk ends and find catenoid limits.

Associated with every quaternionic representation of a compact, connected Lie group there is a Seiberg-Witten equation in dimension three. The moduli spaces of solutions to these equations are typically non-compact. We construct Kuranishi models around boundary points of a partially compactified moduli space. The Haydy…

Unified product Lie groups and their quotient spaces are analyzed for dynamics.

We prove a version of the affine Kempf-Ness theorem for non-algebraic symplectic structures and shifted moment maps, and use it to describe hyperkahler quotients of T*G, where G is a complex reductive group.

We prove that all entire smooth strictly convex self-shrinking solutions on $\mathbb{R}^n$ to the Hessian quotient flows must be quadratic. This generalizes the rigidity theorem for entire self-shrinking solutions to the Lagrangian mean curvature flow in pseudo-Euclidean space due to Ding-Xin \cite{DX}. Moreover, we sh…

4D gradient solitons with constant curvature are rigid.

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

Paper establishes estimates for nonlinear equations on compact manifolds.

Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit a presentation as Lagrangian quotients, and can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable holomorphic vector bundles of fixed rank and degree on a smooth complex projecti…

Second derivative pinching estimates are proved for a class of elliptic and parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply convergence of convex hypersurfaces to spheres under these flows, improving earl…

Generalizes mean-value inequality to orbifold setting.

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

We introduce a complete set of combinatorial data that encode the category $2\mathfrak{Cob}$ of all $2$-cobordisms. As an application, we show that the local monoids of $2\mathfrak{Cob}$ do not have finitely axiomatizable equational theories. As yet another application, we construct a von-Neumann-regular extension of t…