arXiv research

Minimal existence time for Willmore flow established for smooth and weak Lipschitz initial data.

In this paper, we prove the long-time existence and uniqueness of the conical Kähler-Ricci flow with weak initial data which admits $L^{p}$ density for some $p>1$ on Fano manifold. Furthermore, we study the convergence behavior of this kind of flow.

Survey on preserving curvature bounds for non-smooth Ricci flow.

We establish the positive energy theorem for weak asymptotically anti-de Sitter initial data sets with distributional curvature under the weak dominant energy condition.

Study constructs solutions for evolving hypersurfaces using inverse spacetime mean curvature.

Under weak regularity assumptions, only, we develop a fully geometric theory of vacuum Einstein spacetimes with T2 symmetry, establish the global well-posedness of the initial value problem for Einstein's field equations, and investigate the global causal structure of the constructed spacetimes. Our weak regularity ass…

In this paper we propose a class of local definitions of weak lower scalar curvature bounds that is well defined for $C^0$ metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from $C^0$…

In this note, we define and study Kähler-Ricci flow with initial data not being smooth with some natural applications.

In an earlier work joint with X. X. Chen and G. Tian, we introduced the weak Kähler-Ricci flow for various geometric motivations. In the current work, we take further consideration on setting up the weak flow. Namely, the initial class is allowed to be no longer Kähler.

Optimal spectral estimators and AMP combine for efficient weak recovery in orthogonally invariant GLMs.

We propose a weak formulation for the binormal curvature flow of curves in $\R^3.$ This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to hold, as long as self-intersections do not occur. We also prove a global existence t…

Weak base-point freeness leads to Kähler-Ricci flow diameter bounds.

New data shows multiple black holes can form from modified vacuum data.

We consider the problem of evolving hypersurfaces by mean curvature flow in the presence of obstacles, that is domains which the flow is not allowed to enter. In this paper, we treat the case of complete graphs and explain how the approach of M. Saez and the second author yields a global weak solution to the original p…

We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface $\partialΩ_0$, we show that there exists a weak solution to the null mean curvatu…

We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on negatively curved Riemannian manifolds. We show existence of solutions taking as initial condition a finite Radon measure, not necessarily positive. We then establish uniqueness in the class of nonnegative …

Constructs initial data leading to apparent horizons and tests Penrose Inequality.

Study on when smooth Ricci flow remains smooth at the start.

We introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in asymptotically flat Riemannian manifolds. A theory of weak solutions is developed usin…

A lower bound for the ADM mass is established in terms of angular momentum, charge, and horizon area in the context of maximal, axisymmetric initial data for the Einstein-Maxwell equations which satisfy the weak energy condition. If, on the horizon, the given data agree to a certain extent with the associated model Ker…

Proves weak cosmic censorship for a specific Einstein-scalar field system in 2+1 dimensions.

We show existence and uniqueness of very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds satisfying suitable lower bounds on Ricci curvature, with initial data that can grow at infinity at a prescribed rate, that depends crucially on the curvature bounds. The curvature c…

We derive an upper bound on the waiting time for a variational weak solution to Inverse Mean Curvature Flow in $\mathbb{R}^{n+1}$ to become star-shaped. As a consequence, we demonstrate that any connected surface moving by the flow which is not initially a topological sphere develops a singularity or self-intersection …

By making use of the nice behavior of Hawking masses of slices of a weak solution of inverse mean curvature flow in three dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean…

We prove global existence for solutions arising from small initial data for a large class of quasilinear wave equations satisfying the `weak null condition' of Lindblad and Rodnianski, significantly enlarging upon the class of equations for which global existence is known. In addition to the usual weak null condition, …

We study the phase field method for the volume preserving mean curvature flow. Given an initial $C^1$ hypersurface we proved the existence of the weak solution for the volume preserving mean curvature flow via the reaction diffusion equation with a nonlocal term. We also show the monotonicity formula and the density up…

We prove that the limit hypersurfaces of converging curvature flows are stable, if the initial velocity has a weak sign, and give a survey of the existence and regularity results.

We consider localized deformation for initial data sets of the Einstein field equations with the dominant energy condition. Deformation results with the weak inequality need to be handled delicately. We introduce a modified constraint operator to absorb the first order change of the metric in the dominant energy condit…

Neural networks trained with actor-critic algorithms converge to ODEs under weak convergence analysis.

New boundary condition for weak inverse mean curvature flow in bounded domains.

This paper is devoted to the geometric analysis of the incompressible averaged Euler equations on compact Riemannian manifolds with boundary. The equation also coincides with the model for a second-grade non-Newtonian fluid. We study the analytical and geometrical properties of the Lagrangian flow map. We prove existen…

We show how to reduce the general formulation of the mass-angular momentum inequality, for axisymmetric initial data of the Einstein equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. This procedure is based on a certain deformation of the initial data which p…

Concept drift in learning and classification occurs when the statistical properties of either the data features or target change over time; evidence of drift has appeared in search data, medical research, malware, web data, and video. Drift adaptation has not yet been addressed in high dimensional, noisy, low-context d…

Paper presents robust clustering methods for general mixture models.

Weak correlations explain linear dynamics in deep learning models.

Proves higher regularity for anisotropic inverse mean curvature flow.

In this paper, we prove that on a Fano manifold $M$ which admits a Kähler-Ricci soliton $(\om,X)$, if the initial Kähler metric $\om_{\vphi_0}$ is close to $\om$ in some weak sense, then the weak Kähler-Ricci flow exists globally and converges in Cheeger-Gromov sense. Moreover, if $\vphi_0$ is also $K_X$-invariant, the…

Continuous curve evolution depends on initial shape on sphere.

Researchers solve porous medium equation on noncompact manifolds with Ricci curvature.

Self-training improves weak classifiers in mixture models.

We prove a rigidity result for non-negative scalar curvature perturbations of the Euclidean metric on $\mathbb{R^n}$ , which may be regarded as a weak version of the rigidity statement of the positive mass theorem. We prove our result by analyzing long time solutions of Ricci DeTurck flow. As a byproduct in doing so, w…

We study the new geometric flow that was introduced in [11] that evolves a pair of map and (domain) metric in such a way that it changes appropriate initial data into branched minimal immersions. In the present paper we focus on the existence theory as well as the issue of uniqueness of solutions. We establish that a (…

This paper extends the convergence analysis of Langevin Monte Carlo beyond Poincaré inequalities.

Let $B_1$ be the unit open disk in $\Real^2$ and $M$ be a closed Riemannian manifold. In this note, we first prove the uniqueness for weak solutions of the harmonic map heat flow in $H^1([0,T]\times B_1,M)$ whose energy is non-increasing in time, given initial data $u_0\in H^1(B_1,M)$ and boundary data $γ=u_0|_{\partia…

Method approximates Lipschitz domains with smoother shapes.

Study on dynamic curves with elastic energy and spontaneous curvature.

New non-canonical flows found via parabolic Allen-Cahn equations.

Scarcity of labeled data is a bottleneck for supervised learning models. A paradigm that has evolved for dealing with this problem is data programming. An existing data programming paradigm allows human supervision to be provided as a set of discrete labeling functions (LF) that output possibly noisy labels to input in…