arXiv research

Liouville theorems extended to graphs with bounded geometry.

Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.

The study sets limits on heat equation solutions' Hessians on curved spaces.

Study proves upper bounds for solutions on Riemannian manifolds.

Study shows uniform bounds on torsion and curvature for Chern-Ricci flow solutions.

In our previous work we showed that for an ancient solution to the Ricci flow with nonnegative curvature operator, assuming bounded geometry on one time slice, bounded entropy implies noncollapsing on all scales. In this paper we prove the implication in the other direction, that for an ancient solution with bounded no…

Compact mean curvature flow solutions with bounded curvature in high dimensions are constructed.

New study confirms some mean curvature flow solutions have bounded mean curvature.

Sharp bounds found for minimal surface solutions.

We prove global and local upper bounds for the Hessian of log positive solutions of the heat equation on a Riemannian manifold. The metric is either fixed or evolves under the Ricci flow. These upper bounds supplement the well-known global lower bound.

Study finds lower bounds for solutions on Riemannian orbifolds.

Existence and uniqueness of bounded solutions to complex Monge-Ampère flows on Kähler manifolds.

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.

Stable solutions to a specific equation are one-dimensional.

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…

Improved estimator for least squares using random projections achieves smaller error.

In this paper, we study the existence and non-existence result of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics. We can get some useful local …

Paper improves uncertainty quantification in PINNs using error bounds and solution bundles.

Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, has at most one solution in the class of complete riemannian metric with complex se…

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

Study connects curvature bounds to map existence and flow solutions.

The paper constructs ancient solutions to curvature flows in bounded and unbounded regions.

The Ricci flow preserves product structures with instantaneous curvature bounds.

This paper, focusing on the growth rate of the measure, gives pointwise bounds of solutions of eigenvalue equations of the Laplace-Beltrami operator on noncompact Riemannian manifolds.

In this paper, by the method of moving planes, we establish the monotonicity and symmetry properties of convex solutions for Monge-Ampere systems on bounded smooth planar domains.

The Kähler-Ricci flow's singularities are analyzed with bounds and convergence results.

We show uniqueness of classical solutions of the normalised two-dimensional Hamilton-Ricci flow on closed, smooth manifolds for smooth data among solutions satisfying (essentially) only a uniform bound for the Liouville energy and a natural space-time $L^2$-bound for the time derivative of the solution. The result is s…

We show precompactness results for solutions to parabolic fourth order geometric evolution equations. As part of the proof we obtain smoothing estimates for these flows in the presence of a curvature bound, an improvement on prior results which also require a Sobolev constant bound. As consequences of these results we …

We prove that conservation of probability for the free heat semigroup on a Riemannian manifold $M$ (namely stochastic completeness), hence a linear property, is equivalent to uniqueness of positive, bounded solutions to nonlinear evolution equations of fast diffusion type on $M$ of the form $u_t=Δφ(u)$, $φ$ being an ar…

Two ancient solutions to Gauss curvature flow are identified for cylinders.

The paper studies heat kernels on weighted Riemannian manifolds with curvature bounds.

We use a first-order energy quantity to prove a strengthened statement of uniqueness for the Ricci flow. One consequence of this statement is that if a complete solution on a noncompact manifold has uniformly bounded Ricci curvature, then its sectional curvature will remain bounded for a short time if it is bounded ini…

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…

New findings confirm parallels to De Giorgi's conjecture for phase transitions in higher dimensions.

Researchers estimate gradients of solutions to a Finslerian Allen-Cahn equation.

Paper proves inequalities on Hermitian manifolds with applications to bounded solutions.

We prove results on bounded solutions to backward stochastic equations driven by random measures. Those bounded BSDE solutions are then applied to solve different stochastic optimization problems with exponential utility in models where the underlying filtration is noncontinuous. This includes results on portfolio opti…

We prove stability of solutions of the complex Monge-Ampère equation on compact Hermitian manifolds, when the right hand side varies in a bounded set in $L^p, p>1$ and it is bounded away from zero. Such solutions are shown to be Hölder continuous. As an application we extend a recent result of Székelyhidi and Tosatti o…

In this paper we study the Dirichlet problem of translating mean curvature equations over domains in Riemannian manifolds with dimension $n$. Imitating the generalized solution theory of Miranda-Giusti, we define a new conformal area functional and a generalized solution to this Dirichlet problem. The existence of gene…

Study on stability and continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.

We make use of the flexibility of infinite-index solutions to the Allen-Cahn equation to show that, given any compact hypersurface $Σ$ of R^d, with $d\geq 4$, there is a bounded entire solution of the Allen-Cahn equation on R^d whose zero level set has a connected component diffeomorphic (and arbitrarily close) to a re…

Study finds minimum growth rate for surface solutions.

The paper provides generalization bounds for metric learning using neural network embeddings.

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…

In this paper we will give a rigorous proof of the lower bound for the scalar curvature of the standard solution of the Ricci flow conjectured by G. Perelman. We will prove that the scalar curvature $R$ of the standard solution satisfies $R(x,t)\ge C_0/(1-t)\quad\forall x\in\Bbb{R}^3,0\le t<1$, for some constant $C_0>0…

Derives gradient bounds for f-heat equations on manifolds with Bakry-Emery Ricci curvature.

Paper tightens optimization bounds using conformal prediction.

In this paper we study the geometry and the topology of unbounded domains in the Hyperbolic Space $\mathbb{H} ^n$ supporting a bounded positive solution to an overdetermined elliptic problem. Under suitable conditions on the elliptic problem and the behaviour of the bounded solution at infinity, we are able to show tha…