arXiv research

In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized Ricci flow are solutions which exist for all time $t \in [0, \infty)$ with unif…

Proves existence and uniqueness of viscosity solutions to complex Hessian equations on compact Hermitian manifolds.

Study finds obstacles to solutions for specific equations on compact surfaces.

Proves existence and uniqueness of solutions to the Lp Gaussian Minkowski problem.

In this article we study self-gravitating static solutions of the Einstein-ScalarField system in arbitrary dimensions. We discuss the existence and the non-existence of geodesically complete solutions depending on the form of the scalar field potential $V(φ)$, and provide full global geometric estimates when the soluti…

New methods prove existence of rotating shapes moving in space.

We consider a spinorial Yamabe-type problem on open manifolds of bounded geometry. The aim is to study the existence of solutions to the associated Euler-Lagrange-equation. We show that under suitable assumptions such a solution exists. As an application, we prove that existence of a solution implies the conformal Hija…

Paper proves existence of knot solutions for specific equations.

Verified numerics prove existence of a curvature solution with known symmetries.

A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors, we study the existence or non-existence of non-singular solutions of the normalized Ricci flow on 4-manifolds with non-trivial fu…

Paper investigates curvature problems and existence of solutions.

The dual problem of optimal transportation in Lorentz-Finsler geometry is studied. It is shown that in general no solution exists even in the presence of an optimal coupling. Under natural assumptions dual solutions are established. It is further shown that the existence of a dual solution implies that the optimal tran…

This paper deals with the existence of solutions to a class of fourth order nonlinear elliptic equations. The technique used relies on critical points theory. The solutions appeared as critical points of a functional restricted to a suitable manifold.In the case of constant coefficients we obtain the existence of tree …

In this paper, we derive a mean curvature estimate for eternal solutions (including translating solutions) of almost-calibrated Lagrangian mean curvature flow in complex Euclidean space. As a consequence, we show a non-existence result for eternal solutions of almost-calibrated Lagrangian mean curvature flow.

Study finds least-energy nodal solutions to the Yamabe problem on manifolds with boundary.

Proves smooth solution uniqueness and long-term existence for a parabolic equation on a complex manifold.

Proves existence and uniqueness of solutions for a nonlinear equation on Hilbert manifold.

Proves existence of proper solutions for inverse mean curvature flow.

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

The paper proves short-time existence and uniqueness of Ricci flow on Finsler manifolds.

Study on radial solutions of Lane-Emden system on Cartan-Hadamard manifolds.

Existence of translating solutions shown for curve diffusion flow.

We provide a complete study of existence and uniqueness of solutions to the Lichnerowicz equation in general relativity with arbitrary mean curvature.

The paper finds sign-changing solutions for a specific type of elliptic equation.

We consider a class of finite Markov moment problems with arbitrary number of positive and negative branches. We show criteria for the existence and uniqueness of solutions, and we characterize in detail the non-unique solution families. Moreover, we present a constructive algorithm to solve the moment problems numeric…

The paper examines solutions to a specific type of nonlinear equation in a disk, proving existence and uniqueness.

Proves existence of smooth convex solutions to capillary curvature equations.

Here, we study the existence and uniqueness of solutions to the Ricci flow on Finsler surfaces and show short time existence of solutions for such flows. To this purpose, we first study the Finslerian Ricci-DeTurck flow on Finsler surfaces and find a unique short time solution to this flow. Then, we find a solution to …

Existence of solutions to the Heath-Jarrow-Morton equation of the bond market with linear volatility and general Lévy random factor is studied. Conditions for existence and non-existence of solutions in the class of bounded fields are presented. For the existence of solutions the Lévy process should necessarily be with…

The paper proves the existence of infinitely many nodal solutions to a Paneitz-type equation.

Paper proves solutions for deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds.

Proves existence and uniqueness of weak solutions for specific equations.

We recast the Calabi flow in DeGiorgi's language of minimizing movements. We establish the long time existence of minimizing movements for K-energy with arbitrary initial condition. Furthermore we establish some a priori regularity of these solutions, and that sufficiently regular minimizing movements are smooth soluti…

The paper proves the existence of infinite sign-changing solutions to a Hardy-Sobolev equation on Riemannian manifolds.

Study finds weak solutions for complex map flows with optimal lifespan.

The paper is concerned with the problem of existence of solutions for the Heath-Jarrow-Morton equation with linear volatility. Necessary conditions and sufficient conditions for the existence of weak solutions and strong solutions are provided. It is shown that the key role is played by the logarithmic growth condition…

Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, existence of solutions to the so called Hamilton Ricci flow on Finsler spaces is studied and a short time solution is found. To this end the Finslerian Ricci-DeTurck flow on Finsle…

Global solutions and smoothing effects for reaction-diffusion equations on manifolds.

Study proves long-term solutions to a specific equation on hyperKähler manifolds.

Paper finds smooth convex solutions to curvature problem.

We prove the long time existence and uniqueness of solutions to the parabolic Monge-Ampère equation on compact almost Hermitian manifolds. We also show that the normalization of solution converges to a smooth function in $C^{\infty}$ topology as $t\rightarrow\infty$. Up to scaling, the limit function is a solution of t…

Proves existence and uniqueness of solutions for A_n tt*-Toda equations.

Smooth even solutions found for a generalized convex geometry problem.

Paper shows existence of vortex solutions with specific decay properties.

Study connects curvature bounds to map existence and flow solutions.

In this paper, we prove the existence of classical solutions of the Dirichlet problem for a class of quasi-linear elliptic equations on unbounded domains like a cone or a U-type domain. This problem comes from the study of mean curvature flow and its generalization, the flow by powers of mean curvature. Our approach is…

Problems on region choices for knot and link diagrams solved using Alexander numbering.

We provide a probabilistic solution of a not necessarily Markovian control problem with a state constraint by means of a Backward Stochastic Differential Equation (BSDE). The novelty of our solution approach is that the BSDE possesses a singular terminal condition. We prove that a solution of the BSDE exists, thus part…