arXiv subjects

Neshan Wickramasekera

Publications and source records attributed to Neshan Wickramasekera.

At least 19 recordsLinked to original sources

arXiv2026

Analysis of singularities of area-minimizing currents, Part II: a uniform height bound, estimates away from branch points of rapid decay, and uniqueness of tangent cones

This is the second paper in a series developing a new framework for $n$-dimensional area-minimizing rectifiable currents $T$ of codimension $\geq 2$. In the present article we establish a new height estimate for $T$, which says that in a cylinder in the ambient space, the pointwise distance of $T$ to a union of non-intersecting planes is bounded from above, in the interior, \emph{linearly} by the $L^{2}$ height excess of $T$ relative to the same union of planes, whenever appropriate smallness-of-excess conditions are satisfied. We use this estimate and techniques inspired by the works \cite{Sim93}, \cite{Wic14}, \cite{KrumWic2} to establish a decay estimate for $T$ whenever, among other requirements, $T$ is significantly closer to a union of planes meeting along an $(n-2)$-dimensional subspace than to any single plane. Combined with Theorem~1.1 of Part~I, this implies two main results: (a) $T$ has a unique tangent cone at ${\mathcal H}^{n-2}$ a.e.\ point, and (b) the set of singular points of $T$ where $T$, upon scaling, does not decay \emph{rapidly} to a plane is countably $(n-2)$-rectifiable. In particular, concerning \emph{branch points} of $T$, the work here and in \cite{KrumWica} establishes the fact that rapid decay to a unique tangent plane is the generic behaviour, in the sense that at ${\mathcal H}^{n-2}$ a.e.\ branch point, $T$ decays to a unique tangent plane and has \emph{planar frequency} (or the order of contact with the tangent plane) bounded below by $1 + α$ for some fixed $α\in (0, 1)$ depending only on $n$, $m$ and a mass upper bound for $T$; the planar frequency exists, is uniquely defined and is finite by the approximate monotonicity of the (intrinsic) planar frequency function introduced in Part I.

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arXiv2026

Analysis of singularities of area minimizing currents, Part I: planar frequency, branch points of rapid decay, and weak locally uniform approximation

This is the first paper in a series developing a new framework for $n$-dimensional area-minimizing rectifiable currents $T$ of codim. $\geq 2$. Our approach relies on an intrinsic frequency function for $T$, the \emph{planar frequency}, introduced in the present paper. We establish that planar frequency satisfies an approximate monotonicity property, and takes values $\leq 1$ on cones. These properties imply a \emph{decomposition theorem} for the singular set, which (roughly speaking) asserts the following: for any integer $q \geq 2$, the set of density $q$ singularities decomposes as ${\rm sing}_{q} \, T = {\mathcal S} \cup {\mathcal B}$ for disjoint sets ${\mathcal S}$ and ${\mathcal B}$, where: (I) each point $Z \in {\mathcal S}$ has a neighbourhood ${\mathbf B}_{ρ_{Z}}(Z)$ such that about any point $Z^{\prime} \in {\mathbf B}_{ρ_{Z}}(Z) \cap {\rm spt} \, T$ with density $\geq q$ and at any scale $ρ^{\prime} < ρ_{Z}$, $T$ is significantly closer to some non-planar cone than to any plane, and (II) ${\mathcal B}$ is relatively closed in ${\rm sing}_{q} \, T$ and $T$ satisfies a locally uniform estimate along ${\mathcal B}$ implying decay to a unique tangent plane at a rate $o(ρ^{1 + α})$ as the scale $ρ\to 0$, where $α$ is a locally uniform constant. This is central to the more refined analysis in the subsequent papers. The program establishes: (i) uniqueness of tangent cones at ${\mathcal H}^{n-2}$ a.e. point; (ii) singular set decomposition into fintely many disjoint, locally compact, locally $(n-2)$-rectifiable sets (of locally finite measure); (iii) $T$ admits an asymptotic expansion of finite order $> 1$ with remainder estimates at ${\mathcal H}^{n-2}$-a.e. branch point; and (iv) near any branch point satisfying a specific frequency criterion, $T$ is homeomorphic to an $n$-dimensional disk and admits a $C^{1, μ}$ parameterization.

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arXiv2026

Analysis of singularities of area-minimizing currents, Part III: branch points of planar frequency $\neq$ 2, higher order asymptotics, and the local topology

This is the third part in a series of papers developing a new framework to study the local structure of $n$-dimensional area-minimizing rectifiable currents $T$ of codimension $\geq 2$. Parts I and II introduced an intrinsic frequency function for $T$ -- planar frequency -- and used its monotonicity properties, among other things, to establish that ${\mathcal H}^{n-2}$-a.e. branch point is a rapid-decay branch point where the planar frequency is at least $1 + α$. This paper analyses branch points of planar frequency $\neq 2$. It establishes: (1) higher order asymptotics: at ${\mathcal H}^{n-2}$-a.e. such point, the current admits an expansion of finite order $>1$, with precise decay estimates for the remainder term; (2) branch set decomposition: the set of such branch points locally decomposes into finitely many pairwise disjoint, locally $n-2$ rectifiable sets (of locally finite measure); (3) topological control: near any branch point satisfying a specific planar-frequency criterion, the support of $T$ is homeomorphic to an $n$-dimensional disk and admits a $C^{1, μ}$ parametrization. (Classical complex algebraic examples show that when this frequency criterion fails, the current need not be locally homeomorphic to an $n$-disk). The work here (as well as in parts I & II) avoids the use of center manifolds -- a technically demanding foundational component of the classical Almgren framework -- and uses instead intrinsic geometric arguments based on the monotonicity formula for planar frequency. In part IV, a center manifold is utilised to analyse planar frequency 2 points, where the center manifold becomes necessary and geometrically canonical, satisfying additional simplifying properties. Reduced reliance on center manifolds in our framekwork is necessitated by the structural results it establishes for $T$.

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arXiv2026

A Branch Set Stratification for Stationary Varifolds with Epsilon-Regularity

Suppose $\mathcal{V}$ is a class of stationary integral $n$-varifolds in $B^{n+k}_2(0)\subset\mathbb{R}^{n+k}$ which is closed under weak limits, homotheties, rotations, and disjoint decomposition, and suppose that $\mathcal {V}$ satisfies an $ε$-regularity property near planes of (integer) multiplicity $\leq Q\in \{2,3,\dotsc\}$. This last condition, more precisely, requires that there be a constant $ε= ε({\mathcal V}, Q) \in (0, 1)$ such that if $V\in \mathcal{V}$ is, in the unit cylinder ${\mathbb R}^{k} \times B_{1}^{n}(0)$, $ε$-close as varifolds to the plane $\{0\} \times {\mathbb R}^{n}$ taken with multiplicity $\leq Q$ then, in the half-cylinder ${\mathbb R}^{k} \times B_{1/2}^{n}(0)$, $V$ is represented by the graph of a Lipschitz multi-valued function over $B_{1/2}^{n}(0)$ with uniform quantitative estimates of a $C^{1,α}$ nature. For any varifold in such a class $\mathcal{V}$, we prove that the set of branch points with density $\leq Q$ has Hausdorff dimension $\leq n-2$. By choosing suitable $\mathcal{V}$, a direct consequence of this result and the recently established regularity theorems of the second and third authors (one of which being joint with Becker-Kahn) is that if $V$ is a stationary integral $n$-varifold which is either: (a) represented by the graph of a $2$-valued Lipschitz function; or (b) codimension one, stable, and with no classical singularities of density $<Q$, then the Hausdorff dimension of the density $Q$ branch set ($Q=2$ in (a)) is at most $n-2$. Our proof utilises the planar frequency function introduced by the first and third authors in their work on area minimising currents, and thus does not require the Almgren center manifold for the analysis of branch points except in a single, geometrically canonical case where the center manifold satisfies additional simplifying properties.

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arXiv2025

On the Nature of Stationary Integral Varifolds near Multiplicity 2 Planes

We study stationary integral $n$-varifolds $V$ in the unit ball $B_1(0)\subset\mathbb{R}^{n+k}$. Allard's regularity theorem establishes the existence of $ε= ε(n,k)\in (0,1)$ for which if $V$ is $ε$-close (as varifolds) to the plane $P_0 = \{0\}^k\times\mathbb{R}^n$ with multiplicity 1 then, in $B_{1/2}(0)$, $V$ is represented by a single $C^{1,α}$ minimal graph. However, when instead $P_0$ occurs with multiplicity $Q\in \{2,3,\dotsc\}$, simple examples show that this conclusion, now as a multi-valued graph, may fail, even if $V$ corresponds to an area-minimising rectifiable current. In the present work we investigate the structure of such $V$ which are close to planes with multiplicity $Q>1$, focusing primarily on the case $Q=2$. We show that an $ε$-regularity theorem holds when $V$ is close, as a varifold, to $P_0$ with multiplicity $2$, provided $V$ satisfies a certain topological structural condition on the part of its support where the density of $V$ is $<2$. The conclusion then is that, in $B_{1/2}(0)$, $V$ is represented by the graph of a Lipschitz $2$-valued function over $P_0$ with small Lipschitz constant; in fact, the function is $C^{1,α}$ in a precise generalised sense, and satisfies estimates, implying that all tangent cones at singular points in $B_{1/2}(0)$ are unique and comprised of stationary unions of $4$ half-planes (which may form a union of two distinct planes or a single multiplicity $2$ plane). The theorem does not require any additional assumption on the part of $V$ with density $\geq 2$ (which a priori may be a relatively large set in $\mathcal{H}^n$-measure with high topological complexity). As a corollary, we show that our $ε$-regularity theorem applies unconditionally to stationary $2$-valued Lipschitz graphs with arbitrary Lipschitz constant, yielding improved regularity and uniform a priori estimates.

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arXiv2023

A Structure Theory for Stable Codimension 1 Integral Varifolds with Applications to Area Minimising Hypersurfaces mod p

For any $Q\in\{\frac{3}{2},2,\frac{5}{2},3,\dotsc\}$, we establish a structure theory for the class $\mathcal{S}_Q$ of stable codimension 1 stationary integral varifolds admitting no classical singularities of density $<Q$. This theory comprises three main theorems which describe the nature of a varifold $V\in \mathcal{S}_Q$ when: (i) $V$ is close to a flat disk of multiplicity $Q$ (for integer $Q$); (ii) $V$ is close to a flat disk of integer multiplicity $<Q$; and (iii) $V$ is close to a stationary cone with vertex density $Q$ and support the union of 3 or more half-hyperplanes meeting along a common axis. The main new result concerns (i) and gives in particular a description of $V\in \mathcal{S}_Q$ near branch points of density $Q$. Results concerning (ii) and (iii) directly follow from parts of the work [Wic14] (and are reproduced in Part 2). These three theorems, taken with $Q=p/2$, are readily applicable to codimension 1 rectifiable area minimising currents mod $p$ for any integer $p\geq 2$, establishing local structure properties of such a current $T$ as consequences of little, readily checked, information. Specifically, applying case (i) it follows that, for even $p$, if $T$ has one tangent cone at an interior point $y$ equal to an (oriented) hyperplane $P$ of multiplicity $p/2$, then $P$ is the unique tangent cone at $y$, and $T$ near $y$ is given by the graph of a $\frac{p}{2}$-valued function with $C^{1,α}$ regularity in a certain generalised sense. This settles a basic remaining open question in the study of the local structure of such currents near points with planar tangent cones, extending the cases $p=2$ and $p=4$ of the result which have been known since the 1970's from the De Giorgi--Allard regularity theory ([All72]) and the structure theory of White ([Whi79]) respectively.

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arXiv2021

Fine properties of branch point singularities: stationary two-valued graphs and stable minimal hypersurfaces near points of density $< 3$

We study (higher order) asymptotic behaviour near branch points of stationary $n$-dimensional two-valued $C^{1, μ}$ graphs in an open subset of ${\mathbb R}^{n+m}$. Specifically, if $M$ is the graph of a two-valued $C^{1, μ}$ function $u$ on an open subset $Ω\subset {\mathbb R}^{n}$ taking values in the space of un-ordered pairs of points in ${\mathbb R}^{m}$, and if the integral varifold $V = (M, θ),$ where the multiplicity function $θ\, : \, M \rightarrow \{1, 2\}$ is such that $θ=2$ on the set where the two values of $u$ agree and $θ=1$ otherwise, is stationary in $Ω\times {\mathbb R}^{m}$ with respect to the mass functional, we show that at ${\mathcal H}^{n-2}$-a.e.\ point $Z$ along its branch locus $u$ decays asymptotically, modulo its single valued average, to a unique non-zero two-valued cylindrical harmonic tangent function $φ^{(Z)}$ which is homogeneous of some degree $\geq 3/2$. As a corollary, we obtain that the branch locus of $u$ is countably $(n-2)$-rectifiable, and near points $Z$ where the degree of homogeneity of $φ^{(Z)}$ is equal to $3/2$, the branch locus is an embedded real analytic submanifold of dimension $n-2$. These results, combined with the recent works \cite{M} and \cite{MW}, imply a stratification theorem for the (relatively open) set of density $< 3$ points of a stationary codimension 1 integral $n$-varifold with stable regular part and no triple junction singularities.

math.AP↗

arXiv2021

The inhomogeneous Allen--Cahn equation and the existence of prescribed-mean-curvature hypersurfaces

We prove that for any given compact Riemannian manifold $N$ of dimension $n+1 \geq 3$ and any non-negative Lipschitz function $g$ on $N$, there exists a quasi-embedded, boundaryless hypersurface $M \subset N,$ of class $C^{2, α}$ for any $α\in (0,1),$ such that $M$ is the image of a two-sided immersion whose mean curvature is given by $gν$ for an appropriate choice of continuous unit normal $ν$ to the immersion; and moreover, the singular set $Σ= \overline{M} \setminus M$ is empty if $2 \leq n \leq 6,$ finite if $n=7$ and satisfies ${\mathcal H}^{n-7 + γ}(Σ) = 0$ for every $γ>0$ if $n \geq 8$. Here quasi-embedded means that near every non-embedded point, $M$ is the union of two embedded $C^{2, α}$ disks intersecting tangentially with each disk lying on one side of the other. If $g >0$ then $\overline{M}$ is the boundary of a Caccioppoli set. Our proof of this theorem is PDE theoretic and relies, when $g>0$ and $g\in C^{1,1}(N)$, on (i) a mountain pass construction of solutions to the inhomogeneous Allen--Cahn equation and (ii) a regularity result for integral varifolds arising from a Morse-index bounded, energy bounded, sequence of solutions to the (inhomogeneous) Allen--Cahn equation. The case of non-negative Lipschitz $g$ follows by approximation, based on the estimates that we establish.

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arXiv2020

Stable prescribed-mean-curvature integral varifolds of codimension 1: regularity and compactness

In a previous paper we developed a regularity and compactness theory in Euclidean ambient spaces for codimension 1 weakly stable CMC integral varifolds satisfying two (necessary) structural conditions. Here we generalize this theory to the setting where the mean curvature (of the regular part of the varifold) is prescribed by a given $C^{1, α}$ ambient function $g$ and the ambient space is a general $(n+1)$-dimensional Riemannian manifold; we give general conditions that imply that the support of the varifold, away from a possible singular set of dimension $\leq n-7,$ is the image of a $C^{2}$ immersion with continuous unit normal $ν$ and mean curvature $gν$. These conditions also identify a compact class of varifolds subject to an additional uniform mass bound. If $g$ does not vanish anywhere, or more generally if the set $\{g=0\}$ is sufficiently small (e.g. if ${\mathcal H}^{n}\,(\{g=0\})=0$), then the results and their proofs are quite analogous to the CMC case. When $g$ is arbitrary however, a number of additional considerations must be taken into account: first, the correct second variation assumption is that of finite Morse index (rather than weak stability); secondly, to have a geometrically useful theory with compactness conclusions, one of the two structural conditions on the varifold must be weakened; thirdly, in view of easy examples, this weakening of a structural hypothesis necessitates additional hypotheses in order to reach the same conclusion. We identify two sets of additional assumptions under which this conclusion holds for general $g$: one of them gives regularity and compactness, the other one gives regularity and is applicable to certain phase transition problems. We also provide corollaries for multiplicity $1$ varifolds associated to reduced boundaries of Caccioppoli sets. In these cases, some or all of the structural assumptions become redundant.

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arXiv2019

Curvature estimates and sheeting theorems for weakly stable CMC hypersurfaces

Weakly stable constant mean curvature (CMC) hypersurfaces are stable critical points of the area functional with respect to volume preserving deformations. We establish a pointwise curvature estimate (in the non-singular dimensions) and a sheeting theorem (in all dimensions) for weakly stable CMC hypersurfaces, giving an effective version of the compactness theorem for weakly stable CMC hypersurfaces established in the recent work of the first and third-named authors. Our results generalize the curvature estimate and the sheeting theorem proven respectively by Schoen--Simon--Yau and Schoen--Simon for strongly stable hypersurfaces.

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arXiv2018

Stable CMC integral varifolds of codimension $1$: regularity and compactness

We give two structural conditions on a codimension $1$ integral $n$-varifold with first variation locally summable to an exponent $p>n$ that imply the following: whenever each orientable portion of the $C^{1}$-embedded part of the varifold (which is non-empty by the Allard regularity theory) is stationarity and the $C^{2}$-immersed part of it is stable with respect to the area functional for volume preserving deformations, its support, except possibly on a closed set of codimension $7$, is an immersed constant-mean-curvature (cmc) hypersurface of class $C^{2}$ that can fail to be embedded only at points where locally the support is the union of two $C^{2}$ embedded cmc disks with only tangential intersection. Both structural conditions are necessary for the conclusions and involve only those parts of the varifold that are made up of embedded $C^{1, α}$-regular pieces coming together in a regular fashion, making them easy to check in principle. We show also that any family of codimension 1 integral varifolds satisfying these structural and variational hypotheses as well as locally uniform mass and mean curvature bounds is compact in the varifold topology. Our results generalize both the regularity theory of the second author (for stable minimal hypersurfaces) and the regularity theory of Schoen--Simon, for hypersurfaces satisfying a priori a smallness hypothesis on the singular set in addition to the variational hypotheses). Corollaries of the main varifold regularity theorem are obtained for sets of locally finite perimeter, which generalize the regularity theory of Gonzalez--Massari--Tamanini for boundaries that locally minimize perimeter subject to the fixed enclosed volume constraint.

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arXiv2018

Fine properties of branch point singularities: Dirichlet energy minimizing multi-valued functions

In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multi-valued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most $(n-2),$ where $n$ is the dimension of its domain. Almgren used this result in an essential way to show that the same upper bound holds for the dimension of the singular set of an area minimizing $n$-dimensional rectifiable current of arbitrary codimension. In either case, the dimension bound is sharp. We develop estimates to study the asymptotic behaviour of a multi-valued Dirichlet energy minimizer on approach to its singular set. Our estimates imply that a Dirichlet energy minimizer at ${\mathcal H}^{n-2}$ a.e. point of its singular set has a unique set of homogeneous multi-valued cylindrical tangent functions (blow-ups) to which the minimizer, modulo a set of single-valued harmonic functions, decays exponentially fast upon rescaling. A corollary is that the singular set is countably $(n-2)$-rectifiable. Our work is inspired by the work of L. Simon on the analysis of singularities of minimal submanifolds in multiplicity 1 classes, and uses some new estimates and strategies together with techniques from Wickramasekera's prior work to overcome additional difficulties arising from higher multiplicity and low regularity of the minimizers in the presence of branch points. The results described here were announced in earlier work of the authors where the special case of two-valued Dirichlet minimizing functions was treated.

math.AP↗

arXiv2016

The blow up method for Brakke flows: networks near triple junctions

We introduce a parabolic blow-up method to study the asymptotic behavior of an integral Brakke flow of planar networks (i.e. a 1-dimensional integral Brakke flow in a two dimensional region) weakly close in a space-time region to a static multiplicity 1 triple junction $J$. We show that such a network flow is regular in a smaller space-time region, in the sense that it consists of three curves coming smoothly together at a single point at 120 degree angles, staying smoothly close to $J$ and moving smoothly. Using this result and White's stratification theorem, we deduce that whenever an integral Brakke flow of networks in a space-time region ${\mathcal R}$ has no static tangent flow with density $\geq2$, there exists a closed subset $Σ\subset {\mathcal R}$ of parabolic Hausdorff dimension at most 1 such that the flow is classical in ${\mathcal R} \setminus Σ$, i.e. near every point in ${\mathcal R} \setminus Σ$, the flow, if non-empty, consists of either an embedded curve moving smoothly or three embedded curves meeting smoothly at a single point at 120 degree angles and moving smoothly. In particular, such a flow is classical at all times except for a closed set of times of ordinary Hausdorff dimension at most $\frac{1}{2}$.

math.AP↗

arXiv2013

Fine properties of branch point singularities: Two-valued harmonic functions

In the 1980's, Almgren developed a theory of multi-valued Dirichlet energy minimizing functions on $n$ dimensional domains and used it, in an essential way, to bound the Hausdorff dimension of the singular sets of area minimizing rectifiable currents of dimension $n$ and codimension $\geq 2$. Recent work of the second author shows that two-valued $C^{1, μ}$ harmonic functions on $n$ dimensional domains, which are typically-non-minimizing stationary points of Dirichlet energy, play an essential role in the study of multiplicity 2 branch points of stable codimension 1 rectifiable currents of dimension $n$. In all of these cases (of multi-valued harmonic functions and minimal currents), it is known that the branch sets have Hausdorff dimension $\leq n-2.$ In this paper we initiate a study of the local structure of branch sets. We show that the branch set of a two-valued Dirichlet energy minimizing function or a two-valued $C^{1, μ}$ harmonic function, in each closed ball of its domain, is either empty or has positive $(n-2)$-dimensional Hausdorff measure and is equal to the union of a finite number of locally compact, locally $(n-2)$-rectifiable sets. Our method is inspired by the work of L. Simon on the structure of singularities of minimal submanifolds in compact, multiplicity 1 classes.

math.AP↗

arXiv2013

A sharp strong maximum principle and a sharp unique continuation theorem for singular minimal hypersurfaces

We prove the two theorems of the title, settling two long standing questions in the local theory of singular minimal hypersurfaces. The sharpness of either result is with respect to its hypothesis on the size of the allowable singular sets. The proofs of both theorems rely heavily on the author's recent regularity and compactness theory for stable minimal hypersurfaces, and on earlier work of Ilmanen, Simon and Solomon--White.

math.DG↗

arXiv2013

A general regularity theory for stable codimension 1 integral varifolds

We give a necessary and sufficient geometric structural condition for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities; when this condition is satisfied, the singular set is empty if the dimension of the varifold is 6 or smaller, discrete if the dimension is 7 and has Hausdorff codimension at least 7 if the dimension is 8 or larger. No initial smallness assumption on the singular set is necessary for these conclusions. The work in particular settles the long standing question, left open by the Schoen-Simon Regularity Theory, as to which weakest size hypothesis on the singular set guarantees the validity of the above conclusions. An optimal strong maximum principle for stationary codimension 1 integral varifolds follows.

math.DG↗

arXiv2010

A frequency function and singular set bounds for branched minimal immersions

We show that any 2-valued C^{1, α} (α\in (0, 1)) function u = {u_{1}, u_{2}} on an open ball B in {\mathbb R}^{n} with values u_{1}, u_{2} \in {\mathbb R}^{k} whose graph, viewed as a varifold with multiplicity 2 at points where u_{1} = u_{2} and with multiplicity 1 at points where u_{1}, u_{2} are distinct, is stationary in the cylinder B \times {\mathbb R}^{k} must be a C^{1, 1/2} function, and the set of its branch points, if non-empty, must have Hausdorff dimension (n-2) and locally positive (n-2)-dimensional Hausdorff measure. The C^{1, 1/2} regularity is optimal.

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arXiv2010

Stable phase interfaces in the van der Waals--Cahn--Hilliard theory

We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals--Cahn--Hilliard energy functionals with perturbation parameter tending to 0 is supported by an embedded smooth stable minimal hypersurface in low dimensions and an embedded smooth stable minimal hypersurface away from a closed singular set of co-dimension at least 7 in general dimensions. This result was previously known in case the critical points are local minimizers of energy, in which case the limit-hypersurface is locally area minimizing and its (normalized) multiplicity is 1 a.e. Our theorem uses earlier work of the first author establishing stability of the limit-interface as an integral varifold, and relies on a recent general theorem of the second author for its regularity conclusions in the presence of higher multiplicity.

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