Proves existence and uniqueness of viscosity solutions to complex Hessian equations on compact Hermitian manifolds.
problem Existence and uniqueness of viscosity solutions to complex Hessian equations.
method Proves existence and uniqueness using viscosity solutions and determinant domination conditions.
result Viscosity solutions exist and are unique under certain conditions.
Proves existence and uniqueness of solutions for a nonlinear equation on Hilbert manifold.
problem Proving existence and uniqueness of solutions for a nonlinear equation on Hilbert manifold.
method Analyzes the equation on Hilbert manifold, proving existence and uniqueness of solutions.
result Demonstrates that solutions are in the Hilbert manifold and are gradient flows.
Constructs solutions to Einstein-Maxwell-current system using Sasakian manifolds.
problem Solving the Einstein-Maxwell-Current system with inhomogeneous charged particle density.
method Using Sasakian manifolds to specify magnetic field and electric current.
result Solutions with arbitrary function describing charged particle density and curvature.
We establish the estimates of modulus of continuity for viscosity solutions of nonlinear evolution equations on manifolds, extending previous work of B. Andrews and J. Clutterbuck for regular solutions on manifolds \cite{AC3} and the first author's recent work for viscosity solutions in Euclidean spaces \cite{me1}.
In this paper, we study geometric rigidity of Riemannian manifolds admitting stable solutions of certain elliptic problems (stability in a variational sense), that is, under suitable hypotheses, we are able to characterize the Riemannian manifold which admits a stable solution. Furthermore, under the non-negativity of …
Solutions grow for a special type of math problem on curved spaces.
problem Yamabe problem on manifolds with umbilic boundary
method Building blowing-up solutions for a supercritical perturbation
result Existence of solutions for n>7 and non-vanishing Weyl tensor
Survey on Kähler-Ricci flow solutions.
problem Existence of solutions for Kähler-Ricci flow.
method Survey of recent developments.
result Discussion of solutions existing for all positive times.
Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.
problem Existence and uniqueness of solutions to loop equations in generalized Frobenius manifolds.
method Proves existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
result Existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
Study proves long-term solutions to a specific equation on hyperKähler manifolds.
problem Proving long-term existence and uniqueness of solutions to a parabolic quaternionic Monge-Ampère equation.
method Proved long-term existence and uniqueness using parabolic quaternionic Monge-Ampère type equation.
result Solution converges smoothly to the unique solution of the Monge-Ampère equation.
Paper constructs solutions to WDVV equations for Frobenius manifolds.
problem Solving open WDVV equations for Frobenius manifolds.
method Explicit construction of flat F-manifolds and principal hierarchies.
result Recovery of polynomial solutions for A- and D-type singularities.
Solves long-time solutions for a specific equation on hyperkähler manifolds.
problem Finding solutions to a specific equation on hyperkähler manifolds.
method Introduced a parabolic quaternionic Monge-Ampère equation and proved its long-time solvability.
result Smooth convergence to a solution of the quaternionic Monge-Ampère equation.
The paper proves solutions for Yamabe equations on manifolds with boundary.
problem Existence and multiplicity of positive solutions for Yamabe equations.
method Use of isoparametric functions to prove existence and multiplicity results.
result Existence and multiplicity results for positive solutions of Yamabe equations.
Constructing solutions to the heterotic G2 system on specific types of manifolds.
problem Finding solutions to the heterotic G2 system on certain types of manifolds. method Investigating a 1-parameter family of G2-connections on tangent bundles. result Obtaining several approximate and one new class of exact solutions on degenerate 3-(α,δ)-Sasaki manifolds. Study finds least-energy nodal solutions to the Yamabe problem on manifolds with boundary.
problem Existence of sign-changing solutions to the Yamabe problem on manifolds with boundary.
method Variational approach, analysis of conformal invariants, and sharp energy estimates.
result Existence of least-energy nodal solutions when the manifold is positive and the boundary has non-negative constant mean curvature.
Proves smooth solution uniqueness and long-term existence for a parabolic equation on a complex manifold.
problem Existence and uniqueness of solutions to a parabolic equation on compact complex manifolds.
method Uses parabolic Donaldson's equation to prove existence and uniqueness of smooth solutions.
result Smooth solutions to the parabolic Donaldson's equation on compact complex manifolds exist and are unique for all time.
The paper proves short-time existence and uniqueness of Ricci flow on Finsler manifolds.
problem Existence and uniqueness of Ricci flow solutions on Finsler manifolds.
method Investigation of short-time existence and uniqueness of Ricci flow solutions on Finsler manifolds.
result Theorems demonstrating the short-time existence of the flow solution for n-dimensional Finsler manifolds and the uniqueness of the solution for isotropic Finsler manifolds.
Derives Li & Yau estimates for heat equations on manifolds.
problem Analyzing positive solutions of semilinear heat equations on manifolds.
method Adapts Li & Yau estimates to derive new inequalities.
result Derives Harnack inequality and discusses monotonicity, convexity, decay estimates.
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…
Defines and studies solutions to complex equations on Hermitian manifolds.
problem Solving complex equations on Hermitian manifolds.
method Extending recent theories, defines and studies pluripotential solutions to degenerate parabolic complex Monge-Ampère equations.
result Establishes existence and uniqueness of weak Chern-Ricci flow on complex compact varieties with log terminal singularities.
Solve supercritical Yamabe problem on manifolds with non-umbilic boundary.
problem Solving supercritical Yamabe problem on manifolds with non-umbilic boundary.
method Building blowing-up solutions for a supercritical perturbation of the Yamabe problem.
result Constructed solutions for a supercritical perturbation of the Yamabe problem on manifolds with non-umbilic boundary.
Study proves upper bounds for solutions on Riemannian manifolds.
problem Proving upper bounds for solutions of Leibenson's equation on Riemannian manifolds.
method Proved upper bounds equivalent to a euclidean-type Sobolev inequality.
result Upper bounds for solutions of Leibenson's equation on Riemannian manifolds are equivalent to euclidean-type Sobolev inequalities.
In this work we consider viscosity solutions to second order partial differential equations on Riemannian manifolds. We prove maximum principles for solutions to Dirichlet problem on a compact Riemannian manifold with boundary. Using a different method, we generalize maximum principles of Omori and Yau to a viscosity v…
Gradient estimates for solutions to a p-Laplacian equation on Riemannian manifolds.
problem Gradient estimates for positive weak solutions to a p-Laplacian equation on Riemannian manifolds.
method Morser iteration technique
result Gradient estimates show that positive weak solutions do not exist under certain conditions on manifolds with nonnegative Ricci curvature.
Study on semiconcavity of solutions to gradient obstacle problems on compact manifolds.
problem Gradient obstacle problems on compact Riemannian manifolds.
method Uniform semiconcavity estimates and fine convergence results for solutions and free boundaries.
result The elastic and λ-elastic sets of solutions converge to the cut locus and λ-cut locus of the manifold. We prove the uniqueness of solutions of the Ricci flow on complete noncompact manifolds with bounded curvatures using the De Turck approach. As a consequence we obtain a correct proof of the existence of solution of the Ricci harmonic flow on complete noncompact manifolds with bounded curvatures.
The paper studies CR Yamabe solutions on Sasakian manifolds with nonnegative curvature.
problem Characterizing CR Yamabe solutions on Sasakian manifolds with nonnegative curvature.
method Analyzes solutions to the CR Yamabe equation in noncompact (2n+1)-dimensional Sasakian manifolds with nonnegative curvature. result The Heisenberg group H1 is the only (complete) Sasakian space with nonnegative Tanaka-Webster scalar curvature admitting a (nontrivial) positive solution. Stability and Hölder continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.
problem Establishing Hölder continuity of solutions to complex Monge-Ampère equations.
method Stability result for solutions in Lp space, Hölder continuity proof. result Solutions are Hölder continuous with the same exponent as in the Kähler case.
Study reveals how manifold geometry impacts linear regression solutions.
problem Impact of manifold geometry on linear regression solutions.
method Linear regression applied to manifold-structured data, focusing on extrinsic geometry.
result Linear regression does not have a unique solution on flat manifolds.
The eigenvalue problem for the square integrable solutions is studied usually for elliptic equations. In this note we consider such a problem for the hyperbolic Klein-Gordon equation on Lorentzian manifolds. The investigation could help to answer the question why elementary particles have a discrete mass spectrum. An i…
Study finds solutions to flows by negative curvature powers.
problem Curvature flows with negative powers.
method Closed self-similar solutions in warped product manifolds, proving non-strict convexity.
result Proves self-similar solutions are slices of warped product manifolds.
New solutions found for G2 system on squashed manifolds.
problem Finding solutions to the heterotic G2 system on squashed manifolds.
method Constructing families of solutions using squashed metrics on 7-sphere or Aloff-Wallach space.
result Obtained AdS3 solutions for most squashing parameters, excluding a specific case.
The study sets limits on heat equation solutions' Hessians on curved spaces.
problem Bounding Hessians of positive solutions to heat equations on Kähler manifolds.
method Global and local upper bounds for Hessian matrices under curvature constraints.
result Improved bounds on Hessians for Riemannian manifolds with lower sectional curvature.
Study shows uniqueness of solutions on complex manifolds without requiring solution decay.
problem Uniqueness of solutions to Monge-Ampere equation on complex manifolds.
method Caccioppoli inequality techniques applied to Kähler manifolds with sub-quadratic volume growth.
result Uniqueness of bounded C1,1 solutions to Monge-Ampere equation without decay requirement. The paper constructs solutions to a critical Dirac equation on spheres.
problem Solving the critical Dirac equation on spheres with singularities.
method Constructing Delaunay-type solutions and another kind of singular solutions.
result The constructed solutions are building blocks for singular solutions on Spin manifolds.
Study on stability and continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.
problem Stability and continuity of solutions to degenerate complex Monge-Ampère equations.
method Analysis of Hölder continuity and global continuity of solutions.
result Established uniform diameter bound for the twisted Chern-Ricci flow.
Study critical quasilinear equations on Riemannian manifolds with curvature constraints.
problem Investigate critical quasilinear elliptic equations on Riemannian manifolds with nonnegative Ricci curvature.
method Utilize a new nonlinear Kato inequality and Cheng-Yau type gradient estimates for positive solutions.
result Classify positive solutions to the critical p-Laplace equation and show rigidity concerning the ambient manifold. The study finds continuous solutions to complex Hessian equations on compact Hermitian manifolds.
problem Finding continuous solutions to complex Hessian equations on compact Hermitian manifolds.
method Deriving an L∞-estimate for bounded solutions to the complex m-th Hessian equations on compact Hermitian manifolds, assuming a positive right-hand side in the Orlicz space Lmn(logL)n(h∘log∘logL)n. result Establishing the existence of continuous solutions to the complex Hessian equation under the prescribed assumptions.
Noncompact Ricci-flat solutions have infinite unstable dimensions.
problem Understanding unstable dimensions of noncompact Ricci-flat solutions.
method Derived sufficient conditions for infinite-dimensional unstable manifolds.
result Noncompact Ricci-flat solutions have uncountably many unstable perturbations.
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∞ topology as t→∞. Up to scaling, the limit function is a solution of t…
Develops weak PINNs for efficient manifold solutions of hyperbolic equations.
problem Challenges in approximating weak solutions of nonlinear hyperbolic equations on manifolds.
method Introduces a novel weak PINN (wPINN) formulation on manifolds leveraging well-posedness theory.
result Demonstrates efficient approximation of entropy solutions on manifolds with a complexity independent of ambient space dimension.
Study uncoupled solutions to Dirac-Yang-Mills equations on spin manifolds.
problem Condition for vanishing Dirac current on harmonic spinors.
method Perturbation theory and index theorem.
result Existence of uncoupled solutions, classification of connection forms.
The paper proves the existence of infinite sign-changing solutions to a Hardy-Sobolev equation on Riemannian manifolds.
problem Existence of solutions to a specific type of Hardy-Sobolev equation on Riemannian manifolds.
method Addressed using the properties of isoparametric functions and focusing on the distance function from a submanifold.
result Proves the existence of infinite sign-changing solutions to the Hardy-Sobolev equation.
We derive estimates relating the values of a solution at any two points to the distance between the points, for quasilinear isotropic elliptic equations on compact Riemannian manifolds, depending only on dimension and a lower bound for the Ricci curvature. These estimates imply sharp gradient bounds relating the gradie…
The paper classifies solutions to semilinear equations on curved spaces.
problem Classifying solutions to semilinear equations on manifolds with nonnegative Ricci curvature.
method Proving classification results for subcritical and critical semilinear elliptic equations.
result Strong rigidity results for nontrivial solutions in the critical case.
Unique solution found for quaternionic Monge-Ampère equation on specific HKT manifolds.
problem Solving the quaternionic Monge-Ampère equation on HKT manifolds with an HKT foliation.
method Study of quaternionic Monge-Ampère equation on HKT manifolds with specific foliation properties.
result Unique solution for the equation for every basic datum.
Study ancient solutions to free boundary mean curvature flow in convex manifolds.
problem Understanding ancient solutions to free boundary mean curvature flow in convex manifolds.
method Establish rigidity results and construct foliations to describe ancient solutions.
result Ancient solutions to free boundary mean curvature flow are rigid and exhaust all possibilities under certain conditions.
Finite-time extinction and smoothing effects in fractional fast diffusion on manifolds.
problem Finite-time extinction and smoothing effects in fractional fast diffusion equations.
method Nonlinear semigroups techniques, weighted Lp spaces, fractional Green function. result Sharp extinction rates and pointwise lower bounds for solutions.
The reduced thin-sandwich equations (RTSE) appear within Wheeler's thin-sandwich approach towards the Einstein constraint equations (ECE) of general relativity. It is known that these equations cannot be well-posed in general, but, on closed manifolds, sufficient conditions for well-posedness have been established. In …