arXiv research

The paper proves smoothness of stationary varifolds.

Study quantizes energy distribution in inhomogeneous phase transitions.

We provide a measure based topology for certain unions of C2 rectifiable submanifolds of mixed dimensions in Rn. In this topology lower dimensional sets remain in the limit as measures when higher dimensional sets collapse down to them. For example a decreasing sequence of spheres may have a limit consisting of just a …

Study the singular limit of a boundary reaction equation, showing energy concentration and varifold support.

The paper defines and proves the existence of decompositions of integral varifolds.

In this work it is shown that every integral varifold in an open subset of Euclidian space of locally bounded first variation can be covered by a countable collection of submanifolds of class C^2. Moreover, the mean curvature of each member of the collection agrees with the mean curvature of the varifold almost everywh…

Study local minimizers of Ginzburg-Landau functionals in high dimensions, showing energy measures converge to rectifiable measures.

Develops a PDE approach to constructing nontrivial anisotropic surfaces.

We prove that the support of an $m$ dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered $\mathscr{H}^{m}$ almost everywhere by a countable union of $m$ dimensional submanifolds of class $\mathcal{C}^{2}$. We obtain this result using the …

This paper extends Euclidean theorems to anisotropic settings for varifolds.

We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature f…

Paper provides estimates for varifolds with critical mean curvature.

We prove a Poincaré, and a general Sobolev type inequalities for functions with compact support defined on a $k$-rectifiable varifold $V$ defined on a complete Riemannian manifold with positive injectivity radius and sectional curvature bounded above. Our techniques allow us to consider Riemannian manifolds $(M^n,g)$ w…

New varifolds with capillary boundary properties studied.

We develop a suitable generalization of Almgren's theory of varifolds in a lorentzian setting, focusing on area, first variation, rectifiability, compactness and closure issues. Motivated by the asymptotic behaviour of the scaled hyperbolic Ginzburg-Landau equations, and by the presence of singularities in lorentzian m…

Shows smoothness of varifolds with specific boundary angles.

Graphs with bounded anisotropic mean curvature are regular almost everywhere.

The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration by parts identities for certain compositions with smooth functions. In this class the idea of ze…

This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts. Assuming the varifold to s…

We prove existence and partial regularity of integral rectifiable $m$-dimensional varifolds minimizing functionals of the type $\int |H|^p$ and $\int |A|^p$ in a given Riemannian $n$-dimensional manifold $(N,g)$, $2\leq m<n$ and $p>m$, under suitable assumptions on $N$ (in the end of the paper we give many examples of …

The paper studies surfaces in a bounded domain with orthogonal boundaries and proves curvature estimates.

Proves regularity for stable varifolds near specific cones.

If one considers an integral varifold $I^m\subseteq M$ with bounded mean curvature, and if $S^k(I)\equiv\{x\in M: \text{ no tangent cone at $x$ is }k+1\text{-symmetric}\}$ is the standard stratification of the singular set, then it is well known that $\dim S^k\leq k$. In complete generality nothing else is known about …

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

Constructs approximate mean curvature flows for general varifolds.

Study boundary behavior of limit interfaces in Riemannian manifolds without convexity assumptions.

For a given family of smooth closed curves $γ^1,...,γ^α\subset\mathbb{R}^3$ we consider the problem of finding an elastic \emph{connected} compact surface $M$ with boundary $γ=γ^1\cup...\cupγ^α$. This is realized by minimizing the Willmore energy $\mathcal{W}$ on a suitable class of competitors. While the direct minimi…

In this paper we get a version of mean value inequality for generalized self-expander type submanifolds in Euclidean space. As the application, we prove that if mean curvature flow $M(t)$ on the self-expander in Euclidean space subconverges to an $n$-rectifiable varifold $T$ in weak sense for $t$ goes to the singular t…

Study shows non-uniqueness of Brakke flow near flat singular points.

Brakke flow support is parabolically rectifiable

Michael-Simon inequality proven for anisotropic energies close to area.

We use min-max techniques to produce nontrivial solutions $u_ε:M\to \mathbb{R}^2$ of the Ginzburg-Landau equation $Δu_ε+\frac{1}{ε^2}(1-|u_ε|^2)u_ε=0$ on a given compact Riemannian manifold, whose energy grows like $|\logε|$ as $ε\to 0$. When the degree one cohomology $H^1_{dR}(M)=0$, we show that the energy of these s…

New theory for area of Legendrian surfaces, proving smoothness and variational results.

In this paper, we address the problem of orientation that naturally arises when representing shapes like curves or surfaces as currents. In the field of computational anatomy, the framework of currents has indeed proved very efficient to model a wide variety of shapes. However, in such approaches, orientation of shapes…

We study the asymptotics as $p\uparrow 2$ of stationary $p$-harmonic maps $u_p\in W^{1,p}(M,S^1)$ from a compact manifold $M^n$ to $S^1$, satisfying the natural energy growth condition $\int_M|du_p|^p=O(\frac{1}{2-p}).$ Along a subsequence $p_j\to 2$, we show that the singular sets $Sing(u_{p_j})$ converge to the sup…

The study examines stationary integral varifolds near multiplicity 2 planes, proving regularity under specific conditions.

The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that differentials are Borel functions, higher order rectifiability of the set of differentiabili…

The paper establishes structure theory for stable varifolds with applications to area minimising hypersurfaces.

For Ginzburg-Landau vortices, energy quantization holds only when density is less than 2.

New varifold example shows decomposition failure.

We show that metrics that maximize the k-th Steklov eigenvalue on surfaces with boundary arise from free boundary minimal surfaces in the unit ball. We prove several properties of the volumes of these minimal submanifolds. For free boundary minimal submanifolds in the ball we show that the boundary volume is reduced up…

We establish a new estimate for the Ginzburg-Landau energies $E_ε(u)=\int_M\frac{1}{2}|du|^2+\frac{1}{4ε^2}(1-|u|^2)^2$ of complex-valued maps $u$ on a compact, oriented manifold $M$ with $b_1(M)\neq 0$, obtained by decomposing the harmonic component $h_u$ of the one-form $ju:=u^1du^2-u^2du^1$ into an integral and frac…

Upper bound for Morse index of min-max varifolds.

In this paper, we introduce a version of the moving plane method that applies to potentially quite singular hypersurfaces, generalizing the classical moving plane method for smooth hypersurfaces. Loosely speaking, our version for varifolds shows that smoothness and symmetry at infinity (respectively at the boundary) ca…

The paper simplifies arguments for stationary varifolds results.

This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space with its first variation given by either a Radon measure or a function in some Lebesgue space. Pointwise decay results for the quadratic tilt-excess are established for those varifolds. The results are optimal in terms of…

Generalizes Reilly inequality to varifolds and analyzes equality cases.

Counterexample and new proof for curvature varifolds.