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6 votes
2 answers
390 views

Continuity of perimeter with respect to metric

Let $\Omega$ be an open set in a closed manifold, $(M^n, g)$. We can define the perimeter as $$\text{Per}_g(\Omega) = \sup\bigg\{\int_{\Omega} \text{div}_g(T) dVol_g, \; : \; T \in C^1(M, T M), \quad \...
JMK's user avatar
  • 337
5 votes
0 answers
165 views

Singularities of phase interfaces in closed surfaces

Let $(\Sigma,g)$ be a compact surface without boundary. Given $\epsilon > 0$, the $\epsilon$-Allen-Cahn equation is the semilinear elliptic PDE $\epsilon \Delta_g u - \epsilon^{-1} W'(u) = 0$, with ...
Leo Moos's user avatar
  • 5,038
3 votes
0 answers
100 views

Are there Lojasiewicz-Simon estimates with boundary?

Let $M$ be an analytic manifold with boundary $\partial M$, equipped with a Riemannian metric $g$, which is also analytic up to and including the boundary. Are there Lojasiewicz–Simon estimates ...
Leo Moos's user avatar
  • 5,038
2 votes
0 answers
207 views

Tangent cones at infinity and the regularity of minimal submanifolds

In the famous paper by D. Fischer-Colbrie "Some rigidity theorems for minimal submanifolds of the sphere", the very first sentence reads: It is well known that the regularity of minimal ...
Cris.giansu's user avatar