Publications (23)
Decomposition and minimality of Lagrangian submanifolds in nearly Kähler manifolds
Lars Schäfer, Knut Smoczyk
We show that Lagrangian submanifolds in six-dimensional nearly Kähler (non Kähler) manifolds and in twistor spaces $Z\sp{4n+2}$ over quaternionic Kähler manifolds $Q\sp{4n}$ are…
Curvature Decay Estimates of Graphical Mean Curvature Flow in Higher Codimensions
Knut Smoczyk, Mao-Pei Tsui, Mu-Tao Wang
We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions. To the best of our knowledge, this is the first such estimates without assuming s…
A characterization of the grim reaper cylinder
Francisco Martin, Jesus Perez-Garcia, Andreas Savas-Halilaj +1
In this article we prove that a connected and properly embedded translating soliton in with uniformly bounded genus on compact sets which is -asymptotic to two…
The Fermi Flow and its Application to Geometry
Knut Smoczyk
We introduce the notion of Fermi flow for hypersurfaces in Riemannian manifolds. It turns out that this is a powerful tool to study the geometry of distance surfaces about a given…
Local non-collapsing of volume for the Lagrangian mean curvature flow
Knut Smoczyk
We prove an optimal control on the time-dependent measure of a measurable set under a reparametrized Lagrangian mean curvature flow of almost calibrated submanifolds in a Calabi-Ya…
Self-shrinkers of the mean curvature flow in arbitrary codimension
Knut Smoczyk
For hypersurfaces of dimension greater than one, Huisken showed that compact self-shrinkers of the mean curvature flow with positive scalar mean curvature are spheres. We will prov…
Mean curvature flow in higher codimension - Introduction and survey
Knut Smoczyk
In this text we outline the major techniques, concepts and results in mean curvature flow with a focus on higher codimension. In addition we include a few novel results and some ma…
Generalized Lagrangian mean curvature flows in symplectic manifolds
Knut Smoczyk, Mu-Tao Wang
An almost Kähler structure on a symplectic manifold consists of a Riemannian metric and an almost complex structure such that the symplectic form satisfies…
Mean curvature flow with flat normal bundles
Knut Smoczyk, Guofang Wang, Y. L. Xin
We show that flatness of the normal bundle is preserved under the mean curvature flow in the Euclidean space and use this to generalize a classical result for hypersurfaces due to…
Mean Curvature Flows of Lagrangian Submanifolds with Convex Potentials
Knut Smoczyk, Mu-Tao Wang
This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function o…
Evolution of contractions by mean curvature flow
Andreas Savas-Halilaj, Knut Smoczyk
We investigate length decreasing maps between Riemannian manifolds , of dimensions and , respectively. Assuming that is compact and is complet…
Codimension two mean curvature flow of entire graphs
Andreas Savas-Halilaj, Knut Smoczyk
We consider the graphical mean curvature flow of maps , , and derive estimates on the growth rates of the evolved graphs, based on a ne…
The strong elliptic maximum principle for vector bundles and applications to minimal maps
Andreas Savas-Halilaj, Knut Smoczyk
Based on works by Hopf, Weinberger, Hamilton and Evans, we state and prove the strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds and…
Homotopy of area decreasing maps by mean curvature flow
Andreas Savas-Halilaj, Knut Smoczyk
Let be a smooth area decreasing map between two Riemannian manifolds $(M,\gm)$ and $(N,\gn)$. Under weak and natural assumptions on the curvatures of $(M,\gm)$ and $(N,\…
The hyperbolic mean curvature flow
Philippe G. LeFloch, Knut Smoczyk
We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems f…
Self-expanders of the mean curvature flow
Knut Smoczyk
We study self-expanding solutions of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are prod…
Graphical mean curvature flow with bounded bi-Ricci curvature
Renan Assimos, Andreas Savas-Halilaj, Knut Smoczyk
We consider the graphical mean curvature flow of strictly area decreasing maps , where is a compact Riemannian manifold of dimension and a complete Riemanni…
Mean curvature flow of area decreasing maps between Riemann surfaces
Andreas Savas-Halilaj, Knut Smoczyk
In this article we give a complete description of the evolution of an area decreasing map induced by its mean curvature in the situation where and are complete R…
Evolution of spacelike surfaces in anti-De Sitter space by their Lagrangian angle
Knut Smoczyk
We study spacelike hypersurfaces in anti-De Sitter spacetime that evolve by the Lagrangian angle of their Gauà maps.
A canonical way to deform a Lagrangian submanifold
Knut Smoczyk
We derive some important geometric identities for Lagrangian submanifolds immersed in a Kähler manifold and prove that there exists a canonical way to deform a Lagrangian submanif…
Generalized Lagrangian mean curvature flows: the cotangent bundle case
Knut Smoczyk, Mao-Pei Tsui, Mu-Tao Wang
In [SW2], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short ti…
Lagrangian mean curvature flow of Whitney spheres
Andreas Savas-Halilaj, Knut Smoczyk
It is shown that an equivariant Lagrangian sphere with a positivity condition on its Ricci curvature develops a type-II singularity under the Lagrangian mean curvature flow that re…
On the topology of translating solitons of the mean curvature flow
Francisco Martin, Andreas Savas-Halilaj, Knut Smoczyk
In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow.