arXiv subjects

Brian Krummel

Publications and source records attributed to Brian Krummel.

16 recordsLinked to original sources

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 $1 + \alpha$. 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 $C^{1, \mu}$ 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 $T$.

math.DG↗

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 $\epsilon$-regularity property near planes of (integer) multiplicity $\leq Q\in \{2,3,\dotsc\}$. This last condition, more precisely, requires that there be a constant $\epsilon = \epsilon({\mathcal V}, Q) \in (0, 1)$ such that if $V\in \mathcal{V}$ is, in the unit cylinder ${\mathbb R}^{k} \times B_{1}^{n}(0)$, $\epsilon$-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 $C^{1,\alpha}$ 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.

math.DG↗

arXiv2023

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 $1 + \alpha$ for some fixed $\alpha \in (0, 1)$ 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.

math.DG↗

arXiv2023

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 ${\mathbf B}_{\rho_{Z}}(Z)$ such that about any point $Z^{\prime} \in {\mathbf B}_{\rho_{Z}}(Z) \cap {\rm spt} \, T$ with density $\geq q$ and at any scale $\rho^{\prime} < \rho_{Z}$, $T$ 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 $o(\rho^{1 + \alpha})$ as the scale $\rho \to 0$, where $\alpha$ 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 $C^{1, \mu}$ parameterization.

math.DG↗

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↗

arXiv2020

Existence and regularity results for the penalized thin obstacle problem with variable coefficients

In this paper we give a comprehensive treatment of a two-penalty boundary obstacle problem for a divergence form elliptic operator, motivated by applications to fluid dynamics and thermics. Specifically, we prove existence, uniqueness and optimal regularity of solutions, and establish structural properties of the free boundary. The proofs are based on tailor-made monotonicity formulas of Almgren, Weiss, and Monneau-type, combined with the classical theory of oblique derivative problems.

math.AP↗

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↗

arXiv2017

Higher codimension relative isoperimetric inequality outside a convex set

We consider an isoperimetric inequality for $(m+1)$-dimensional area minimizing submanifolds of arbitrary codimension which lie outside a convex set $\mathcal{K} \subset \mathbb{R}^{n+1}$ and are bounded by a submanifold of $\mathbb{R}^{n+1} \setminus \mathcal{K}$ and the convex set $\mathcal{K}$. We show that the least value of the isoperimetric ratio is attained for an $(m+1)$-dimensional flat half-disk of $\mathbb{R}^{n+1}_+$. This extends prior work of Choe, Ghomi, and Ritoré in codimension one and proves a conjecture of Choe in the case of relative area minimizers.

math.OC↗

arXiv2017

On the regularity of the free boundary in the $p$-Laplacian obstacle problem

We study the regularity of the free boundary in the obstacle for the $p$-Laplacian, $\min\bigl\{-Δ_p u,\,u-φ\bigr\}=0$ in $Ω\subset\mathbb R^n$. Here, $Δ_p u=\textrm{div}\bigl(|\nabla u|^{p-2}\nabla u\bigr)$, and $p\in(1,2)\cup(2,\infty)$. Near those free boundary points where $\nabla φ\neq0$, the operator $Δ_p$ is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when $\nabla φ=0$ then $Δ_p$ is singular or degenerate, and nothing was known about the regularity of the free boundary at those points. Here we study the regularity of the free boundary where $\nabla φ=0$. On the one hand, for every $p\neq2$ we construct explicit global $2$-homogeneous solutions to the $p$-Laplacian obstacle problem whose free boundaries have a corner at the origin. In particular, we show that the free boundary is in general not $C^1$ at points where $\nabla φ=0$. On the other hand, under the "concavity" assumption $|\nabla φ|^{2-p}Δ_p φ<0$, we show the free boundary is countably $(n-1)$-rectifiable and we prove a nondegeneracy property for $u$ at all free boundary points.

math.AP↗

arXiv2016

Regularity of minimal submanifolds and mean curvature flows with a common free boundary

Let $N$ be a smooth $(n+l)$-dimensional Riemannian manifold. We show that if $V$ is an area-stationary union of three or more $C^{1,μ}$ $n$-dimensional submanifolds-with-boundary $M_k \subset N$ with a common boundary $Γ$, then $Γ$ is smooth and each $M_k$ is smooth up to $Γ$ (real-analytic in the case $N$ is real-analytic). This extends a previous result of the author for codimension $l = 1$. We additionally show that if $\{V_t\}_{t \in (-1,1)}$ is a Brakke flow such that each time-slice $V_t$ is a union of three or more $n$-dimensional submanifolds-with-boundary $M_{k,t} \subset N$ with a common boundary $Γ_t$ and with parabolic $C^{2+μ}$ regularity in time-space, then $\{Γ_t\}_{t \in (-1,1)}$ and $\{M_{k,t}\}_{t \in (-1,1)}$ are smooth (second Gevrey with real-analytic time-slices in the case $N$ is real-analytic).

math.DG↗

arXiv2016

Isoperimetry with upper mean curvature bounds and sharp stability estimates

It was proved by Almgren that among boundaries whose mean curvature is bounded from above, perimeter is uniquely minimized by balls. We obtain sharp stability estimates for Almgren's isoperimetric principle and, as an application, we deduce a sharp description of boundaries with almost constant mean curvature under a total perimeter bound which prevents bubbling.

math.OC↗

arXiv2014

Constant frequency and the higher regularity of branch sets

We consider a two-valued function $u$ that is either Dirichlet energy minimizing, $C^{1,μ}$ harmonic, or in $C^{1,μ}$ with an area-stationary graph such that Almgren's frequency (restricted to the singular set) is continuous at a singular point $Y_0$. As a corollary of recent work of Wickramasekera and the author, if the frequency of $u$ at $Y_0$ equals $1/2+k$ for some integer $k \geq 0$, then the singular set of $u$ is a $C^{1,τ}$ submanifold and we have estimates on the asymptotic behavior of $u$ at singular points. Using a nontrivial modification of the argument of Wickramasekera and author, we show that the frequency of $u$ at $Y_0$ cannot equal an integer and therefore must equal $1/2+k$ for some integer $k \geq 0$. We then use the asymptotic behavior of $u$ and partial Legendre-type transformations based on those of Kinderlehrer, Nirenberg, and Spruck to show that the singular set in this case is in fact real analytic.

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

Existence and regularity of multivalued solutions to elliptic equations and systems

We extend the work of Simon and Wickramasekera, who constructed a large class of $C^{1,μ}$ multivalued solutions to the minimal surface equation, to produce $C^{1,μ}$ multivalued solutions to more general classes of elliptic equations and systems, including the minimal surface system with small boundary data and the Laplace equation. We use methods for differential equations, which are more general than the specific minimal submanifold approach adopted by Simon and Wickramasekera. We also prove the branch set of the graphs of the solutions constructed by Simon and Wickramasekera are real analytic submanifolds by inductively using Schauder estimates.

math.DG↗

arXiv2013

Regularity of minimal hypersurfaces with a common free boundary

Let $N$ be a Riemannian manifold and consider a stationary union of three or more $C^{1,μ}$ hypersurfaces-with-boundary $M_k$ in $N$ with a common boundary $Γ$. We show that if $N$ is smooth, then $Γ$ is smooth and each $M_k$ is smooth up to $Γ$ (real analytic in the case $N$ is real analytic). Consequently we strengthen a result of Wickramasekera to conclude that under the stronger hypothesis that $V$ is a stationary, stable, integral $n$-varifold in an $(n+1)$-dimensional, smooth (real analytic) Riemannian manifold such that the support of $\|V\|$ is nowhere locally the union of three or more smooth (real analytic) hypersurfaces-with-boundary meeting along a common boundary, the singular set of $V$ is empty if $n = 6$, is discrete if $n = 7$, and has Hausdorff dimension at most $n-7$ if $n \geq 8$.

math.DG↗

arXiv2011

Classification of Einstein metrics on the product of an interval with a three-sphere

We present a complete classification of Einstein metrics on the space M = I \times S^3, where I is the interval (0,l) or (0,\infty) or their closures, and we consider separate metric functions f and h (functions of I) for the base and fiber of the Hopf fibration S^1 -> S^3 -> S^2. All such metrics yielding smooth and complete manifolds are included and discussed. The results are surprisingly rich, including many well-known examples and several one-parameter families of metrics with a variety of geometries.

math.DG↗