arXiv research

Generalizes O'Neill's equations to pseudo-Finsler submersions.

The fundamental equations of Gauss, Codazzi and Ricci provide the conditions for local isometric embeddability. In general, the three fundamental equations are independent for surfaces in Riemannian 4-manifolds. In contrast, we prove in this article that for arbitrary Lorentz surfaces in Lorentzian Kaehler surfaces the…

In this paper we will prove a maximum principle for the solutions of linear parabolic equation on complete non-compact manifolds with a time varying metric. We will prove the convergence of the Neumann Green function of the conjugate heat equation for the Ricci flow in $B_k\times (0,T)$ to the minimal fundamental solut…

Compatibility equations adapted to magnetic geometry.

In this note we show the convergence of the fundamental solutions of the parabolic equations assuming the Cheeger-Gromov convergence of the underlying manifolds and the uniform $L^1$-bound of the solutions. We also prove a local integral estimate of fundamental solutions.

For a fundamental solution of Laplace's equation on the $R$-radius $d$-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geo…

Due to the isotropy $d$-dimensional hyperbolic space, there exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. On the $R$-radius hyperboloid model of $d$-dimensional hyperbolic geometry with $R>0$ and $d\ge 2$, we compute azimuthal Fourier expansions for a fundamental so…

In this paper we construct a parametrix for the forward fundamental solution of the wave and Klein-Gordon equations on asymptotically de Sitter spaces without caustics. We use this parametrix to obtain asymptotic expansions for solutions of the inhomogeneous equation and to obtain a uniform L^p estimate for a family of…

We find fundamental solutions to p-Laplace equations with drift terms in the Heisenberg group and Grushin-type planes. These solutions are natural generalizations to the fundamental solutions discovered by Beals, Gaveau, and Greiner for the Laplace equation with drift term. Our results are independent of the results of…

The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface $Z$ in an asymptotically simple spacetime satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-…

Fundamental solutions found for PDEs in Finsler geometry.

The present paper describes a relation between the quotient of the fundamental group of a smooth quasi-projective variety by its second commutator and the existence of maps to orbifold curves. It extends previously studied cases when the target was a smooth curve. In the case when the quasi-projective variety is a comp…

Study concavity of solutions to elliptic equations under conformal deformations.

Unified treatment of two extension problems using heat equation in Heisenberg group.

The paper decomposes spacelike hypersurface properties for general relativistic vacuum equations.

Study shows the second fundamental form of pseudospherical surfaces is universal and not dependent on specific solutions.

The classical Cartan's structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. In order to study the mathematical properties of singularities, we need to study the geometry of manifolds endowed on the tangent b…

Let $(M,g(t))$, $0\le t\le T$, $\partial M\neφ$, be a compact $n$-dimensional manifold, $n\ge 2$, with metric $g(t)$ evolving by the Ricci flow such that the second fundamental form of $\partial M$ with respect to the unit outward normal of $\partial M$ is uniformly bounded below on $\partial M\times [0,T]$. We will pr…

We show that for a representation of the fundamental group of a triangulated closed 3-manifold (not necessarily hyperbolic) into $\PSL$ so that any edge loop has non-trivial image under the representation, there exist uncountably many solutions to the hyperbolic gluing equation whose associated representations are conj…

We briefly recall a fundamental exterior differential system introduced by the author and then apply it to the case of three dimensions. Here we find new global tensors and intrinsic invariants of oriented Riemaniann 3-manifolds. The system leads to a remarkable Weingarten type equation for surfaces on hyperbolic 3-spa…

Solves complex Monge-Ampère equations on Kähler manifolds.

We prove that if the fundamental 4-form of an almost-quaternionic Hermitian manifold (M, Q, g) of dimension at least eight satisfies the conformal-Killing equation, then (M, Q, g) is quaternionic-Kahler.

Decides undecidability of equations and first-order theory for Seifert 3-manifold groups.

Survey explores cohomology's roles in applied math and sciences.

Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial differential equations, are presented, which arise naturally in some longstanding, …

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of type I $κ$-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman's pr…

Giving a new form of the vortex mode equation by a proper change of parameter, our aim is to analyze the point and contact symmetries of the new equation. Fundamental invariants and a form of general solutions of point transformations along with some specific examples are also derived.

The Poisson equation on manifolds plays an fundamental role in many applications. Recently, we proposed a novel numerical method called the Point Integral method (PIM) to solve the Poisson equations on manifolds from point clouds. In this paper, we prove the convergence of the point integral method for solving the Pois…

Study variational problems for integral invariants of maps between pseudo-Riemannian manifolds.

We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of potential flat torsionless submanifolds. We show that all potential flat…

We describe the asymptotic behavior of Palais-Smale sequences associated to certain Yamabe-type equations on manifolds with boundary. We prove that each of those sequences converges to a solution of the limit equation plus a finite number of "bubbles" which are obtained by rescaling fundamental solutions of the corresp…

In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's …

In this paper, we derive a Sobolev inequality along an extended Ricci flow and prove a point-wise Guassian type bound for the fundamental solutions of the conjugate heat equation under the flow.

The paper defines and calculates fourth fundamental form and i-th curvatures for hypersurfaces in 4D Euclidean space.

Enhances neural operators with physics knowledge for more accurate simulations.

In the spirit of Klein's Erlangen Program, we investigate the geometric and algebraic structure of fundamental line complexes and the underlying privileged discrete integrable system for the minors of a matrix which constitute associated Plücker coordinates. Particular emphasis is put on the restriction to Lie circle g…

The study introduces canonical coordinates for Lorentz surfaces and proves a Bonnet-type theorem.

Investigates $J$-equation on holomorphic vector bundles over Kähler manifolds.

The paper applies generalised geometry to semi-Riemannian immersions and hypersurfaces.

The article presents a general discrete time dividend valuation model when the dividend growth rate is a general continuous variable. The main assumption is that the dividend growth rate follows a discrete time semi-Markov chain with measurable space. The paper furnishes sufficient conditions that assure finiteness of …

We consider the class of evolution equations that describe pseudo-spherical surfaces of the form u\_t = F (u, $\partial$u/$\partial$x, ..., $\partial$^k u/$\partial$x^k), k $\ge$ 2 classified by Chern-Tenenblat. This class of equations is characterized by the property that to each solution of a differential equation wi…

The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and non-local) and the presence associated linear problems. It thus contains many important known examples…

We find the fundamental solution to the p-Laplace equation in a class of Hörmander vector fields that generate neither a Carnot group nor a Grushin-type space. The singularity occurs at the sub-Riemannian points which naturally corresponds to finding the fundamental solution of a generalized operator in Euclidean space…

We study the interplay between geometry and partial differential equations. We show how the fundamental ideas we use require the ability to correctly calculate the dimensions of spaces associated to the varieties of zeros of the symbols of those differential equations. This brings to the center of the analysis several …

The Degasperis-Procesi equation's solutions define pseudospherical metrics and can lead to surface collapse.

We find the complete set of fundamental invariants for systems of ordinary differential equations of order $\ge 4$ under the group of point transformations generalizing similar results for contact invariants of a single ODE and point invariants of systems of the second and the third order. It turns out that starting fr…

We show that an equivariantly embedded Hermitian symmetric space in a projective space, which contains neither a projective space nor a hyperquadric as a component, is characterized by their fundamental forms as a local submanifold of the projective space. Using some invariant-theoretic properties of the fundamental fo…

We introduce a class of potential submanifolds in pseudo-Euclidean spaces (each N-dimensional potential submanifold is a special flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space) and prove that each N-dimensional Frobenius manifold can be locally represented as an N-dimensional potential submanif…