arXiv research

The paper finds the Finsler structure of Apollonian weak metric on unit disc.

We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity.

Proves curvature tensor convergence for smoothable spaces.

Study proves existence of weak mean curvature flow with contact angle.

Proves inextendibility of weak null singularities from curvature blow-up.

We deal with a notion of weak binormal and weak principal normal for non-smooth curves of the Euclidean space with finite total curvature and total absolute torsion. By means of piecewise linear methods, we first introduce the analogous notation for polygonal curves, where the polarity property is exploited, and then m…

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

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

Surveying Ricci flow for weak lower scalar curvature bounds.

Study shows how weak inverse anisotropic mean curvature flow behaves at infinity.

Preserves scalar curvature bounds under weak convergence of 3-manifolds.

The study examines conditions for weak nearly cosymplectic manifolds to split into products.

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

Smooth flows with surgery approximate weak mean curvature flows with spherical and neck-pinch singularities.

Proves existence of proper solutions for inverse mean curvature flow.

The paper establishes bounds on scalar curvature on asymptotically flat manifolds.

Proves higher regularity for anisotropic inverse mean curvature flow.

Alternative proof of weak solutions to mean curvature flow using minimizing movements.

Novel weak solutions for volume-preserving mean curvature flow established.

We give a proof that Brakke's mean curvature flow under the unit density assumption is smooth almost everywhere in space-time. More generally, if the velocity is equal in a weak sense to its mean curvature plus some given α-Hölder continuous vector field, then we show C^{2,α} regularity almost everywhere.

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 study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove the global existence of the weak solutions for the mean curvature flow with the terms, by using the modified Allen-Cahn equation that holds useful properties such as the monotonicity formula.

We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general…

New varifold solutions for mean curvature flow converge and are unique.

Study weak quasi contact metric manifolds to generalize K-contact and Sasakian manifolds criteria.

Paper proves mass theorems for nonnegative scalar curvature metrics.

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…

In this paper, we investigate special curves on a weak r-helix submanifold in Euclidean n-space E^{n}. Also, we give the important relations between weak r-helix submanifolds and the special curves such as line of curvature, asymptotic curve and helix line.

Study curvature properties of w.a. S-manifolds with new conditions.

Gradient estimates for solutions to a p-Laplacian equation on Riemannian manifolds.

Paper investigates conditions for independence of weak gradients on metric spaces.

Defines weak normals for irregular curves in high-dimensional spaces.

Develops weak formulation for spacelike flows in pseudo-Euclidean space.

The paper studies a new structure on contact manifolds and its foliation properties.

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…

Long time existence and convergence to a circle is proved for radial graph solutions to a mean curvature type curve flow in warped product surfaces (under a weak assumption on the warp potential of the surface). This curvature flow preserves the area enclosed by the evolving curve, and this fact is used to prove a gene…

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…

In this paper we generalize the Local Removable Singularity Theorem in [16] for minimal laminations to the case of weak $H$-laminations (with $H\in \mathbb{R}$ constant) in a punctured ball of a Riemannian three-manifold. We also obtain a curvature estimate for any weak CMC foliation (with possibly varying constant mea…

New varifold example shows decomposition failure.

We review recent work on the Einstein equations of general relativity when the curvature is defined in a weak sense. Weakly regular spacetimes are constructed, in which impulsive gravitational waves, as well as shock waves, propagate.

In this paper we prove an extrinsic one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature which is independent of the value of the constant mean curvature. We apply this extrinsic one-sided curvature estimate in [24] to prove to prove a weak chord arc type result for these disks…

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…

The problem of prescribing conformally the scalar curvature of a closed Riemannian manifold as a given Morse function reduces to solving an elliptic partial differential equation with critical Sobolev exponent. Two ways of attacking this problem consist in subcritical approximations or negative pseudo gradient flows. W…

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 assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a …

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 $C^0$…

The paper proves existence and growth estimates for inverse mean curvature flow and related $p$-Laplacian Green kernel decay.

We give a new proof of Brakke's partial regularity theorem up to C^{1,ς} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and an…