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# On the mean curvature flow associated to a density

## Vicente Miquel Universidad de Valencia

I’ll describe what is the mean curvature flow associated to a density and will give some account of my recent work with F. Viñado-Lereu.
First, in $\mathbb{R}^n$ with a density $e^\psi$, we study the mean curvature flow associated to the density ($\psi$MCF) of a hypersurface. The main results of this part of the talk concern with the description of the evolution under $\psi$MCF of a closed embedded curve in the plane with a radial density, and with a statement of subconvergence to a $\psi$-minimal closed curve in a surface under some general circumstances.
Second, we define Type I singularities for the $\psi$MCF and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from the part of the $\psi$-curvature due to the density. We describe a family of curves whose evolution under $\psi$MCF (in a Riemannian surface of non-negative curvature with a density which is singular at a geodesic of the surface) produces only type I singularities and study the limits of their blow-ups.
These results and their proofs are collected in:
Miquel, Vicente; Viñado-Lereu, Francisco; "The curve shortening problem associated to a density". Calc. Var. Partial Differential Equations 55 (2016), no. 3, 55:61 and
Miquel, Vicente; Viñado-Lereu, Francisco; "Type I singularities in the curve shortening flow associated to a density” arXiv:1607.08402

# Tres observaciones en Análisis Geométrico

## Vicente Miquel Universidad de Valencia

Primera observación: el primer valor propio de Dirichlet de un tubo alrededor de una subvariedad compleja de $\mathbb{C}P^n$ está acotado por una función del radio y de los grados de los polinomios que definen el centro del tubo. (Trabajo conjunto con M.Carmen Domingo-Juan). Segunda: existen ejemplos de superficies “mean-convex” que, al evolucionar por el flujo por la curvatura media conservando el volumen dejan de ser “mean-convex”. (Trabajo conjunto con Esther Cabezas-Rivas). Tercera: se dan ejemplos de superficies lagrangianas de $\mathbb{C}^2$ que, por el flujo por la curvatura media, se contraen a un punto con la forma de un toro de Clifford. (Trabajo conjunto con Ildefonso Castro y Ana Lerma).

# On the role of Killing vector fields for a good behavior of mean curvature flow

## Vicente Miquel Universidad de Valencia

We shall indicate the rough idea that a hypersurface transversal to a Killing vector field that flows by MCF remains transverse to it and will study this evolution in some warped products of the real line times a riemannian manifold. This talk contains joint work with A. Borisenko.

# Mean curvature flow of graphs in warped products

## Vicente Miquel Universidad de Valencia

Let $M$ be a complete Riemannian manifold which either is compact or has a pole, and let $varphi$ be a positive smooth function on $M$. In the warped product $M times_varphi mathbb R$, we study the flow by the mean curvature of a locally Lipschitz continuous graph on $M$ and prove that the flow exists for all time and that the evolving hypersurface is $C^infty$ for $t>0$ and is a graph for all $t$. Moreover, under certain conditions, the flow has a well defined limit.
Joint work with A. Borisenko

Seminario de Matemáticas. 1ª Planta