6,869 research outputs found
On the entropy of closed hypersurfaces and singular self-shrinkers
Self-shrinkers are the special solutions of mean curvature flow in
that evolve by shrinking homothetically; they serve as
singularity models for the flow. The entropy of a hypersurface introduced by
Colding-Minicozzi is a Lyapunov functional for the mean curvature flow, and is
fundamental to their theory of generic mean curvature flow.
In this paper we prove that a conjecture of Colding-Ilmanen-Minicozzi-White,
namely that any closed hypersurface in has entropy at least
that of the round sphere, holds in any dimension . This result had
previously been established for the cases by Bernstein-Wang using a
carefully constructed weak flow.
The main technical result of this paper is an extension of
Colding-Minicozzi's classification of entropy-stable self-shrinkers to the
singular setting. In particular, we show that any entropy-stable self-shrinker
whose singular set satisfies Wickramasekera's -structural hypothesis
must be a round cylinder .Comment: 35 pages, comments welcome
Moving-centre monotonicity formulae for minimal submanifolds and related equations
Monotonicity formulae play a crucial role for many geometric PDEs, especially
for their regularity theories. For minimal submanifolds in a Euclidean ball,
the classical monotonicity formula implies that if such a submanifold passes
through the centre of the ball, then its area is at least that of the
equatorial disk. Recently Brendle and Hung proved a sharp area bound for
minimal submanifolds when the prescribed point is not the centre of the ball,
which resolved a conjecture of Alexander, Hoffman and Osserman. Their proof
involves asymptotic analysis of an ingeniously chosen vector field, and the
divergence theorem.
In this article we prove a sharp `moving-centre' monotonicity formula for
minimal submanifolds, which implies the aforementioned area bound. We also
describe similar moving-centre monotonicity formulae for stationary
-harmonic maps, mean curvature flow and the harmonic map heat flow.Comment: 20 pages; added reference
Minimal hypersurfaces with small first eigenvalue in manifolds of positive Ricci curvature
In this paper we exhibit deformations of the hemisphere , , for which the ambient Ricci curvature lower bound and
the minimality of the boundary are preserved, but the first Laplace eigenvalue
of the boundary decreases. The existence of these metrics suggests that any
resolution of Yau's conjecture on the first eigenvalue of minimal hypersurfaces
in spheres would likely need to consider more geometric data than a Ricci
curvature lower bound.Comment: 30 pages; changes to sections 3.2, 3.3, typos fixed; to appear in J.
Topol. Ana
Min-max theory for constant mean curvature hypersurfaces
In this paper, we develop a min-max theory for the construction of constant
mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary
closed manifold. As a corollary, we prove the existence of a nontrivial,
smooth, closed, almost embedded, CMC hypersurface of any given mean curvature
. Moreover, if is nonzero then our min-max solution always has
multiplicity one.Comment: 32 pages. More backgrounds and references added. Comments welcome
On the rigidity of mean convex self-shrinkers
Self-shrinkers model singularities of the mean curvature flow; they are
defined as the special solutions that contract homothetically under the flow.
Colding-Ilmanen-Minicozzi showed that cylindrical self-shrinkers are rigid in a
strong sense - that is, any self-shrinker that is mean convex with uniformly
bounded curvature on a large, but compact, set must be a round cylinder. Using
this result, Colding and Minicozzi were able to establish uniqueness of blowups
at cylindrical singularities, and provide a detailed description of the
singular set of generic mean curvature flows.
In this paper, we show that the bounded curvature assumption is unnecessary
for the rigidity of the cylinder if either n is at most 6, or if the mean
curvature is bounded below by a positive constant. These results follow from
curvature estimates that we prove for strictly mean convex self-shrinkers. We
also obtain a rigidity theorem in all dimensions for graphical self-shrinkers,
and curvature estimates for translators of the mean curvature flow.Comment: 15 pages, comments welcome
Min-max theory for networks of constant geodesic curvature
We prove that on a closed surface, for any , our min-max theory for
prescribing mean curvature produces a solution given by a curve of constant
geodesic curvature which is almost embedded, except for finitely many
points, at which the solution is a stationary junction with integer density.
Moreover, each smooth segment has multiplicity one. The key is a classification
of blowups which is new even for .Comment: 11 pages, minor revisio
Rigidity and Curvature Estimates for Graphical Self-shrinkers
Self-shrinkers are hypersurfaces that shrink homothetically under mean
curvature flow; these solitons model the singularities of the flow. It it
presently known that an entire self-shrinking graph must be a hyperplane. In
this paper we show that the hyperplane is rigid in an even stronger sense,
namely: For , any smooth, complete self-shrinker
that is graphical inside a large, but
compact, set must be a hyperplane. In fact, this rigidity holds within a larger
class of almost stable self-shrinkers.
A key component of this paper is the procurement of linear curvature
estimates for almost stable shrinkers, and it is this step that is responsible
for the restriction on . Our methods also yield uniform curvature bounds for
translating solitons of the mean curvature flow.Comment: 20 page
A Decentralized Multi-block ADMM for Demand-side Primary Frequency Control using Local Frequency Measurements
We consider demand-side primary frequency control in the power grid provided
by smart and flexible loads: loads change consumption to match generation and
help the grid while minimizing disutility for consumers incurred by consumption
changes. The dual formulation of this problem has been solved previously by
Zhao et al. in a decentralized manner for consumer disutilities that are twice
continuously differentiable with respect to consumption changes. In this work,
we propose a decentralized multi-block
alternating-direction-method-of-multipliers (DM-ADMM) algorithm to solve this
problem. In contrast to the dual ascent algorithm of Zhao et al., the proposed
DM-ADMM algorithm does not require the disutilities to be continuously
differentiable; this allows disutility functions that model consumer behavior
that may be quite common. In this work, we prove convergence of the DM-ADMM
algorithm in the deterministic setting (i.e., when loads may estimate the
consumption-generation mismatch from frequency measurements exactly). We test
the performance of the DM-ADMM algorithm in simulations, and we compare (when
applicable) with the previously proposed solution for the dual formulation. We
also present numerical results for a previously proposed ADMM algorithm, whose
results were not previously reported
Stochastic Security: Adversarial Defense Using Long-Run Dynamics of Energy-Based Models
The vulnerability of deep networks to adversarial attacks is a central
problem for deep learning from the perspective of both cognition and security.
The current most successful defense method is to train a classifier using
adversarial images created during learning. Another defense approach involves
transformation or purification of the original input to remove adversarial
signals before the image is classified. We focus on defending naturally-trained
classifiers using Markov Chain Monte Carlo (MCMC) sampling with an Energy-Based
Model (EBM) for adversarial purification. In contrast to adversarial training,
our approach is intended to secure pre-existing and highly vulnerable
classifiers.
The memoryless behavior of long-run MCMC sampling will eventually remove
adversarial signals, while metastable behavior preserves consistent appearance
of MCMC samples after many steps to allow accurate long-run prediction.
Balancing these factors can lead to effective purification and robust
classification. We evaluate adversarial defense with an EBM using the strongest
known attacks against purification. Our contributions are 1) an improved method
for training EBM's with realistic long-run MCMC samples, 2) an
Expectation-Over-Transformation (EOT) defense that resolves theoretical
ambiguities for stochastic defenses and from which the EOT attack naturally
follows, and 3) state-of-the-art adversarial defense for naturally-trained
classifiers and competitive defense compared to adversarially-trained
classifiers on Cifar-10, SVHN, and Cifar-100. Code and pre-trained models are
available at https://github.com/point0bar1/ebm-defense.Comment: ICLR 202
Homogenisation of a Row of Dislocation Dipoles from Discrete Dislocation Dynamics
Conventional discrete-to-continuum approaches have seen their limitation in
describing the collective behaviour of the multi-polar configurations of
dislocations, which are widely observed in crystalline materials. The reason is
that dislocation dipoles, which play an important role in determining the
mechanical properties of crystals, often get smeared out when traditional
homogenisation methods are applied. To address such difficulties, the
collective behaviour of a row of dislocation dipoles is studied by using
matched asymptotic techniques. The discrete-to-continuum transition is
facilitated by introducing two field variables respectively describing the
dislocation pair density potential and the dislocation pair width. It is found
that the dislocation pair width evolves much faster than the pair density. Such
hierarchy in evolution time scales enables us to describe the dislocation
dynamics at the coarse-grained level by an evolution equation for the slowly
varying variable (the pair density) coupled with an equilibrium equation for
the fast varying variable (the pair width). The time-scale separation method
adopted here paves a way for properly incorporating dipole-like (zero net
Burgers vector but non-vanishing) dislocation structures, known as the
statistically stored dislocations (SSDs) into macroscopic models of crystal
plasticity in three dimensions. Moreover, the natural transition between
different equilibrium patterns found here may also shed light on understanding
the emergence of the persistent slip bands (PSBs) in fatigue metals induced by
cyclic loads
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