arXiv research

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

The study explores special Ricci-Hessian equations on Kähler manifolds and identifies three types of solutions.

On a manifold of dimension at least six, let $(g,τ)$ be a pair consisting of a Kähler metric g which is locally Kähler irreducible, and a nonconstant smooth function $τ$. Off the zero set of $τ$, if the metric $\hat{g}=g/τ^2$ is a gradient Ricci soliton which has soliton function $1/τ$, we show that $\hat{g}$ is Kähler…

Proves triviality and nonexistence of gradient Ricci solitons as warped metrics.

In this paper we study a Ricci-Hessian type manifold $(\Bbb{M},g,\varphi,f,λ)$ which is closely related to the construction of almost Ricci soliton realized as a warped product. We classify certain classes of the Ricci-Hessian type manifolds and derive some implications for almost Ricci solitons and generalized $m$--qu…

We present the necessary and sufficient conditions for constructing gradient Ricci almost solitons that are realized as warped products. This will be done by means of Bishop-O'Neill's formulas and a particular study of Riemannian manifolds satisfying a Ricci-Hessian type equation. We prove existence results and give an…

In this paper, we study and partially classify those Riemannian man-ifolds carrying a non-identically vanishing function f whose Hessian is minus f times the Ricci-tensor of the manifold.

Compact Riemannian manifolds with mostly positive curvature have finite fundamental groups.

The paper generalizes the Bott-Virasoro group and derives new Euler equations.

Proves solvability of general inverse σ_k equations with constant coefficients.

We present a contact transformation of the generalized Hunter--Saxton equation to the Euler--Poisson equation with special values of the Ovsiannikov invariants. We also find the general solution for the generalized Hunter--Saxton equation.

The paper generalizes Monge-Ampère equations and their solutions in differential geometry.

Study generalizes Hermitian-Einstein equation for cyclic Higgs bundles, proving existence and inequality.

Solves generalized Kazdan-Warner equations on foliated manifolds.

In this work we present a new approach on studying dynamical systems. Combining the two ways of expressing the uncertainty, using probabilistic theory and credibility theory, we have research the generalized fractional hybrid equations. We have introduced the concepts of generalized fractional Wiener process, generaliz…

We study the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge-Ampere (class 6-6), Goursat (class 6-7) and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence method to study the gen…

Proves convexity of level sets of general inverse σ_k equations.

We give a description of recently introduced Doubrov-Ferapontov general heavenly equation in terms of closed differential Plücker two-form, rationally depending on the spectral parameter. We demonstrate that general heavenly equation is an important generating equation in the context of Takasaki hyper-Kähler hierarchy,…

Solves Dirichlet problem for generalized Hitchin's equation on cyclic Higgs bundles.

Generalizes Hodge correlators using quantum master equation concepts.

In this paper, we study the solvability of a general class of fully nonlinear curvature equations, which can be viewed as generalizations of the equations for Christoffel-Minkowski problem in convex geometry. We will also study the Dirichlet problem of the corresponding degenerate equations as an extension of the equat…

We analyze a generalized version of the Black-Scholes equation depending on a parameter $a\!\in \!(-\infty,0)$. It satisfies the martingale condition and coincides with the Black-Scholes equation in the limit case $a\nearrow 0$. We show that the generalized equation is exactly solvable in terms of Hermite polynomials a…

The paper introduces new functionals and equations for complex vector bundles.

New proof shows solutions to Lichnerowicz equation exist.

Generalized diffusion type equations are considered and point symmetry analysis is applied to them. The equations with extremal order point symmetry algebras are described. Some old geometrical results are rederived in connection with theory of these equation.

Solves a generalized Monge-Ampère equation on Kähler surfaces, proving a conjecture.

Geometrically interprets two equations, showing their equivalence and providing solutions.

Growth of spinors in 4D and 3D generalized Seiberg-Witten equations.

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

Study geometric singular solutions of generalized Monge-Ampère equations.

Derives equations of motion for systems with angular momentum on Finsler geometries.

We show that classical Wilczynski--Se-ashi invariants of linear systems of ordinary differential equations are generalized in a natural way to contact invariants of non-linear ODEs. We explore geometric structures associated with equations that have vanishing generalized Wilczynski invariants and establish relationship…

Motion of curves and surfaces in $\R^3$ lead to nonlinear evolution equations which are often integrable. They are also intimately connected to the dynamics of spin chains in the continuum limit and integrable soliton systems through geometric and gauge symmetric connections/equivalence. Here we point out the fact that…

Proves existence and uniqueness of weak solutions for specific equations.

We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The ot…

This paper classifies symmetries of biharmonic heat equations on surfaces of revolution.

Paper solves Hessian equations on Kähler manifolds.

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

The paper introduces new equations in Kähler geometry and proves their solutions and convexity.

Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.

We associate an integrable generalized complex structure to each 2-dimensional symplectic Monge-Ampère equation of divergent type and, using the Gualtieri $\bar{\partial}$ operator, we characterize the conservation laws and the generating function of such equation as generalized holomorphic objects.

Scientific documents rely on both mathematics and text to communicate ideas. Inspired by the topical correspondence between mathematical equations and word contexts observed in scientific texts, we propose a novel topic model that jointly generates mathematical equations and their surrounding text (TopicEq). Using an e…

In this paper, a type of integrable evolution equation--the generalized Landau-Lifshitz equation into $S^n$ is considered. We deal with this equation from a geometric point of view by rewriting it in a geometric form. Through the geometric energy method, we show the global well-posedness of the corresponding Cauchy pro…

StarNet trains deep models without gradients using linear equations.

The paper explores solutions to the distributional Bellman equation in reinforcement learning.

Paper transforms a complex equation into simpler forms for analysis.

Introduces a new 2C extension of the heavenly equation.

The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up …