On the rate of convergence of the $p$-curve shortening flow
In this paper we give rates of convergence for the $p$-curve shortening flow for $p\geq 1$ an integer, which improves on the known estimates and which are probably sharp.
math.AP↗
arXiv subjects
Alexander Murcia
Publications and source records attributed to Alexander Murcia.
The Ricci flow on a cylinder
In this paper we study the Ricci flow on surfaces homeomorphic to a cylinder (that is, a product of the circle with a compact interval). We prove longtime existence results, results on the asymptotic behavior of the flow, and we report on an interesting phenomenon: convergence to constant curvature in the normalised flow,under certain assumptions on the initial data, cannot be exponential.
math.DG↗
The Ricci flow on surfaces with boundary
We show for a non homogeneous boundary value problem for the Ricci flow on the disk that when the initial metric has positive curvature and the boundary is convex then the initial metric is deformed, via the normalized flow and along sequences of times, to a metric of constant curvature and totally geodesic boundary. We also show that when the geodesic curvature of the boundary is nonpositive and the metric is rotationally symmetric, the normalized version of the flow exists for all time.
math.DG↗
On the stability of $m$-fold circles and the dynamics of generalized curve shortening flows
In this paper we study the (asymptotic and exponential) stability of the $m$-fold circle as a solution of the $p$-curve shortening flow ($p\geq 1$ an integer).
math.AP↗