arXiv subjects

Valentina Wheeler

Publications and source records attributed to Valentina Wheeler.

4 recordsLinked to original sources

arXiv2026

Concentration-compactness for the geometric polyharmonic heat flow

We develop a concentration-compactness theory for geometric evolution equations of arbitrarily high order, using the geometric polyharmonic heat flow of closed immersed surfaces in $\R^3$ as the model case. The flow is the $(2p+2)$-order normal evolution \[ \partial_t f=(-1)^{p+1}\Delta^p H\,\nu,\qquad p\geq1, \] which includes the surface diffusion flow when $p=1$. We prove localised energy estimates with sharp cut-off bookkeeping, interior estimates, a lifespan/concentration alternative, tracefree-curvature $\varepsilon$-regularity estimates, and a gap theorem for stationary solutions. These tools are then combined with a blowup argument, the preservation of signed enclosed volume, and the monotonicity of area to rule out singularities below a small tracefree-curvature threshold. Consequently, for connected initial immersions satisfying $\|A^o\|_2^2<\varepsilon$, where $\varepsilon>0$ depends only on the order of the flow, the solution exists for all time and converges exponentially in $C^\infty$ to a round sphere with the preserved enclosed volume.

math.AP↗

arXiv2026

Homogeneous Sobolev gradient flow of the length functional

The well-known curve shortening flow can be formulated as the gradient flow of the length functional on the space of immersed closed planar curves, where the gradient is taken with respect to a reparametrisation-invariant $L^2$ Riemannian metric. This metric is degenerate, giving a geodesic distance of zero between any two curves. We instead consider a family of Sobolev $H^1$ metrics depending on two parameters $\lambda>0$ and $a\in \mathbb R$, where $\lambda$ sets the weight of the first-derivative term, and $a$ indexes a length normalisation which ensures that the metric is scale-homogeneous. For each such metric, the gradient of length can be written explicitly in terms of a convolution with respect to normalised arc length against the periodic Green's function of $(\lambda^2 \partial_x^2-1)$. The associated evolution is a reparametrisation invariant nonlocal ODE whose right-hand side is well-defined even on curves that are not immersed. Working in the optimal low-regularity setting $W^{1,1}(\mathbb S,\mathbb R^2)$, we prove local well-posedness using the Picard--Lindel\"of theorem and convergence to constant maps in finite time when $a<2$, and as $t\to\infty$ when $a\geq 2$. This behaviour is exhibited by round circles, which evolve self-similarly and collapse at an explicit time. We further prove that if the initial curve is an immersion, $C^1$, $C^2$, or bounds a strictly convex set, then each of these properties is preserved along the flow.

math.DG↗

arXiv2024

A curvature flow that deforms curves to an embedded target

In this paper we introduce the target flow -- a specific curve shortening flow with an ambient forcing term -- that, given an embedded (not necessarily convex) target curve, will attempt to evolve a given source curve to that target. The motivation for this flow is to address a question of Yau. Our main result is that the target flow with uniformly normal graphical data converges smoothly to the target, broadening the class of known sources and targets such that Yau's problem has a solution.

math.DG↗

arXiv2023

A Sobolev gradient flow for the area-normalised Dirichlet energy of $H^1$ maps

In this article we study the $H^1(du)$-gradient flow for the energy $E[X] = Q[X]/A[X]$ where $Q[X]$ is the Dirichlet energy of $X$, $A[X]$ is the signedenclosed area of $X$, and $X:\mathbb{S}\rightarrow\mathbb{R}^2$ is a $H^1(du)$ map. We prove that solutions with initially positive signed enclosed area exist eternally, and converge as $t\rightarrow\infty$ to a (possibly multiply-covered) circle. In this way we recover a parametrised isoperimetric inequality for $H^1(du)$ maps.

math.DG↗