arXiv subjects

Tapio Behrndt

Publications and source records attributed to Tapio Behrndt.

3 recordsLinked to original sources

arXiv2014

On the Cauchy problem for the heat equation on Riemannian manifolds with conical singularities

We study the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. We introduce weighted Hölder and Sobolev spaces with discrete asymptotics and we prove existence and maximal regularity of solutions to the Cauchy problem for the inhomogeneous heat equation, when the free term lies in a weighted parabolic Hölder or Sobolev space with discrete asymptotics. This generalizes a result previously obtained by Coriasco, Schrohe, and Seiler (Thm. 7.2 in Math. Z. 244 (2003), 235--269) by different means.

math.AP↗

arXiv2014

Mean curvature flow of Lagrangian submanifolds with isolated conical singularities

In this paper we study the short time existence problem for the (generalized) Lagrangian mean curvature flow in (almost) Calabi--Yau manifolds when the initial Lagrangian submanifold has isolated conical singularities modelled on stable special Lagrangian cones. Given a Lagrangian submanifold $F_0:L\rightarrow M$ in an almost Calabi--Yau manifold $M$ with isolated conical singularities at $x_1,...,x_n\in M$ modelled on stable special Lagrangian cones $C_1,...,C_n$ in $\mathbb{C}^m$, we show that for a short time there exist one-parameter families of points $x_1(t),... x_n(t)\in M$ and a one parameter family of Lagrangian submanifolds $F(t,\cdot):L\rightarrow M$ with isolated conical singularities at $x_1(t),...,x_n(t)\in M$ modelled on $C_1,...,C_n$, which evolves by (generalized) Lagrangian mean curvature flow with initial condition $F_0:L\rightarrow M$.

math.DG↗

arXiv2011

Generalized Lagrangian mean curvature flow in Kähler manifolds that are almost Einstein

We introduce the notion of Kähler manifolds that are almost Einstein and we define a generalized mean curvature vector field along submanifolds in them. We prove that Lagrangian submanifolds remain Lagrangian, when deformed in direction of the generalized mean curvature vector field. For a Kähler manifold that is almost Einstein, and which in addition has a trivial canonical bundle, we show that the generalized mean curvature vector field of a Lagrangian submanifold is the dual vector field associated to the Lagrangian angle.

math.DG↗