156 research outputs found

### Higher regularity of the inverse mean curvature flow

We prove higher regularity properties of inverse mean curvature flow in Euclidean space: A sharp lower bound for the mean curvature is derived for star-shaped surfaces, independently of the initial mean curvature. It is also shown that solutions to the inverse mean curvature flow are smooth if the mean curvature is bounded from below. As a consequence we show that weak solutions of the inverse mean curvature flow are smooth for large times, beginning from the first time where a surface in the evolution is star-shaped

### Topological Change in Mean Convex Mean Curvature Flow

Consider the mean curvature flow of an (n+1)-dimensional, compact, mean
convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We
prove that elements of the m-th homotopy group of the complementary region can
die only if there is a shrinking S^k x R^(n-k) singularity for some k less than
or equal to m. We also prove that for each m from 1 to n, there is a nonempty
open set of compact, mean convex regions K in R^(n+1) with smooth boundary for
which the resulting mean curvature flow has a shrinking S^m x R^(n-m)
singularity.Comment: 19 pages. This version includes a new section proving that certain
kinds of mean curvature flow singularities persist under arbitrary small
perturbations of the initial surface. Newest update (Oct 2013) fixes some
bibliographic reference

### The inverse mean curvature flow and the Riemannian Penrose Inequality

LetM be an asymptotically flat 3-manifold of nonnegative scalar curvature. The Riemannian Penrose Inequality states that the area of an outermost minimal surface N in M is bounded by the ADM mass m according to the formula |N| ≤ 16πm2. We develop a theory of weak solutions of the inverse mean curvature flow, and employ it to prove this inequality for each connected component of N using Geroch’s monotonicity formula for the ADM mass. Our method also proves positivity of Bartnik’s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry

### Convergence of the Allen-Cahn equation with Neumann boundary conditions

We study a singular limit problem of the Allen-Cahn equation with Neumann
boundary conditions and general initial data of uniformly bounded energy. We
prove that the time-parametrized family of limit energy measures is Brakke's
mean curvature flow with a generalized right angle condition on the boundary.Comment: 26 pages, 1 figur

### Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres

Using mixed analytical and numerical methods we investigate the development
of singularities in the heat flow for corotational harmonic maps from the
$d$-dimensional sphere to itself for $3\leq d\leq 6$. By gluing together
shrinking and expanding asymptotically self-similar solutions we construct
global weak solutions which are smooth everywhere except for a sequence of
times $T_1<T_2<...<T_k<\infty$ at which there occurs the type I blow-up at one
of the poles of the sphere. We show that in the generic case the continuation
beyond blow-up is unique, the topological degree of the map changes by one at
each blow-up time $T_i$, and eventually the solution comes to rest at the zero
energy constant map.Comment: 24 pages, 8 figures, minor corrections, matches published versio

### Gradient flows and instantons at a Lifshitz point

I provide a broad framework to embed gradient flow equations in
non-relativistic field theory models that exhibit anisotropic scaling. The
prime example is the heat equation arising from a Lifshitz scalar field theory;
other examples include the Allen-Cahn equation that models the evolution of
phase boundaries. Then, I review recent results reported in arXiv:1002.0062
describing instantons of Horava-Lifshitz gravity as eternal solutions of
certain geometric flow equations on 3-manifolds. These instanton solutions are
in general chiral when the anisotropic scaling exponent is z=3. Some general
connections with the Onsager-Machlup theory of non-equilibrium processes are
also briefly discussed in this context. Thus, theories of Lifshitz type in d+1
dimensions can be used as off-shell toy models for dynamical vacuum selection
of relativistic field theories in d dimensions.Comment: 19 pages, 1 figure, contribution to conference proceedings (NEB14);
minor typos corrected in v

### Investigating Off-shell Stability of Anti-de Sitter Space in String Theory

We propose an investigation of stability of vacua in string theory by
studying their stability with respect to a (suitable) world-sheet
renormalization group (RG) flow. We prove geometric stability of (Euclidean)
anti-de Sitter (AdS) space (i.e., $\mathbf{H}^n$) with respect to the simplest
RG flow in closed string theory, the Ricci flow. AdS space is not a fixed point
of Ricci flow. We therefore choose an appropriate flow for which it is a fixed
point, prove a linear stability result for AdS space with respect to this flow,
and then show this implies its geometric stability with respect to Ricci flow.
The techniques used can be generalized to RG flows involving other fields. We
also discuss tools from the mathematics of geometric flows that can be used to
study stability of string vacua.Comment: 29 pages, references added in this version to appear in Classical and
Quantum Gravit

### Tightness for a stochastic Allen--Cahn equation

We study an Allen-Cahn equation perturbed by a multiplicative stochastic
noise which is white in time and correlated in space. Formally this equation
approximates a stochastically forced mean curvature flow. We derive uniform
energy bounds and prove tightness of of solutions in the sharp interface limit,
and show convergence to phase-indicator functions.Comment: 27 pages, final Version to appear in "Stochastic Partial Differential
Equations: Analysis and Computations". In Version 4, Proposition 6.3 is new.
It replaces and simplifies the old propositions 6.4-6.

### Symmetry of Traveling Wave Solutions to the Allen-Cahn Equation in \Er^2

In this paper, we prove even symmetry of monotone traveling wave solutions to
the balanced Allen-Cahn equation in the entire plane. Related results for the
unbalanced Allen-Cahn equation are also discussed

### Critical behavior of collapsing surfaces

We consider the mean curvature evolution of rotationally symmetric surfaces.
Using numerical methods, we detect critical behavior at the threshold of
singularity formation resembling the one of gravitational collapse. In
particular, the mean curvature simulation of a one-parameter family of initial
data reveals the existence of a critical initial surface that develops a
degenerate neckpinch. The limiting flow of the Type II singularity is
accurately modeled by the rotationally symmetric translating soliton.Comment: 23 pages, 10 figure

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