List of publications

   
   
36 Curvature decay estimates for mean curvature flow in higher codimensions.
Smoczyk, Knut; Tsui, Mao-Pei; Wang, Mu-Tao
Abstract. 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 smallness of first derivatives of the defining map. An immediate application is a convergence theorem of the mean curvature flow of the graph of an area decreasing map between flat Riemann surfaces.
Journal. arXiv:1401.4154
Preprint preprint
35 Evolution of contractions by mean curvature flow.
Savas-Halilaj, Andreas; Smoczyk, Knut
Abstract. We investigate length decreasing maps f between Riemannian manifolds M, N. Assuming that M is compact and N is complete such that M and N satisfy suitable curvature conditions we obtain an explicit estimate for the mean curvature and show that length decreasing maps converge to constant maps. This is a continuation of earlier work on area decreasing maps but here we are able to drop the compactness assumption for the target manifold.
Journal. arXiv:1312.0783.
Preprint preprint
34 Homotopy of area decreasing maps by mean curvature flow.
Savas-Halilaj, Andreas; Smoczyk, Knut
Abstract. Let f be a smooth area decreasing map between two Riemannian manifolds M, N. Under weak and natural assumptions on the curvatures of M and N we prove that the mean curvature flow provides a smooth homotopy of f to a constant map.
Journal. Advances in Mathematics; 2014 Vol 255; 455--473.
Preprint preprint  Article article
33 Bernstein theorems for length and area decreasing minimal maps.
Savas-Halilaj, Andreas; Smoczyk, Knut
Abstract. 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 give some applications in Differential Geometry. Moreover, we use this maximum principle to obtain various rigidity theorems and Bernstein type theorems in higher codimension for minimal maps between Riemannian manifolds.
Journal. Calculus of Variations (to appear).
Preprint preprint  Article article
32 Evolution of spacelike surfaces in anti-De Sitter space by their Lagrangian angle.
Smoczyk, Knut
Abstract. We study spacelike hypersurfaces M in an anti-De Sitter spacetime N of constant sectional curvature -k<0 that evolve by the Lagrangian angle of their Gauss maps. In the two dimensional case we prove a convergence result to a maximal space-like surface, if the Gauss curvature K of the initial surface M satisfies -k<K<k.
Journal. Mathematische Annalen; 2013, Vol 355; no. 4; 1443--1468.
Preprint preprint  Article article
31 Mean curvature flow in higher codimension - Introduction and survey.
Smoczyk, Knut
Abstract. 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 material that cannot be found elsewhere.
Journal. Bär, Christian; Lohkamp, Joachim; Schwarz, Matthias (eds.), Global Differential Geometry, Springer Proceedings in Mathematics, 2012, Volume 17, Part 2, 231-274.
Preprint preprint  Article article
30 Generalized Lagrangian mean curvature flows in symplectic manifolds.
Smoczyk, Knut; Wang, Mu-Tao
Abstract. An almost Kähler structure on a symplectic manifold (N, ω) consists of a Riemannian metric g and an almost complex structure J such that the symplectic form w satisfies ω(·, ·) = g(J(·), ·). Any symplectic manifold admits an almost Kähler structure and we refer to (N, ω, g, J) as an almost Kähler manifold. In this article, we propose a natural evolution equation to investigate the deformation of Lagrangian submanifolds in almost Kähler manifolds. A metric and complex connection on TN defines a generalized mean curvature vector field along any Lagrangian submanifold M of N. We study the evolution of M along this vector field, which turns out to be a Lagrangian deformation, as long as the connection satisfies an Einstein condition. This can be viewed as a generalization of the classical Lagrangian mean curvature flow in Kähler-Einstein manifolds where the connection is the Levi-Civita connection of g. Our result applies to the important case of Lagrangian submanifolds in a cotangent bundle equipped with the canonical almost Kähler structure and to other generalization of Lagrangian mean curvature flows, such as the flow considered by Behrndt in Kähler manifolds that are almost Einstein.
Journal. Asian J. Math.; 2011; Vol 15; no. 1; 129--140.
Preprint preprint  Article article
29 Mean curvature flow of space-like Lagrangian submanifolds in almost para-Kähler manifolds.
Chursin, Mykhaylo; Schäfer, Lars; Smoczyk, Knut
Abstract. Given an almost para-Kähler manifold equipped with a metric and para-complex connection, we define a generalized second fundamental form and generalized mean curvature vector of space-like Lagrangian submanifolds. We then show that the deformation induced by this variant of the mean curvature vector field preserves the Lagrangian condition, if the connection satisfies also some Einstein condition. In case the almost para-Kähler structure is integrable, the flow coincides with the classical mean curvature flow in the pseudo-Riemannian context.
Journal. Calculus of Variations; 2011; Vol 41; no. 1-2; 111--125.
Preprint preprint  Article article
28 Complex and Differential Geometry - Conference held at Leibniz Universität Hannover, September 14 – 18, 2009.
Ebeling, Wolfgang; Hulek, Klaus; Smoczyk, Knut (eds.)
Abstract. This volume contains the Proceedings of the conference "Complex and Differential Geometry 2009", held at Leibniz Universität Hannover, September 14 - 18, 2009. It was the aim of this conference to bring specialists from differential geometry and (complex) algebraic geometry together and to discuss new developments in and the interaction between these fields. Correspondingly, the articles in this book cover a wide area of topics, ranging from topics in (classical) algebraic geometry through complex geometry, including (holomorphic) symplectic and poisson geometry, to differential geometry (with an emphasis on curvature flows) and topology.
Journal. Springer Proceedings in Mathematics, Volume 8, 2011, Springer Verlag.
  Book bok
27 On algebraic selfsimilar solutions of the mean curvature flow.
Smoczyk, Knut
Abstract. In this short note we show that in codimension one all homogeneous algebraic selfsimilar solutions of the mean curvature flow are either algebraic minimal hypersurfaces or belong to the class of the well known self-shrinking quadrics.
Journal. Analysis (Munich); 2011; Vol. 31; 91--102.
Preprint preprint  Article article
26 Curve Shortening on Sasaki Manifolds and the Weinstein Conjecture.
Smoczyk, Knut
Abstract. We propose a new geometric evolution equation for closed curves on contact manifolds. Our flow is a parabolic evolution equation similar to the well known curve shortening flow (mean curvature flow) and is the negative gradient flow of the Reeb energy of closed curves. We analyze the longtime and singular behavior of the flow and describe relations to the Weinstein conjecture on the existence of closed Reeb orbits.
Journal. East-West J. Math.; 2010, Special Vol., 292--305.
Preprint preprint
25 The Sasaki-Ricci flow.
Smoczyk, Knut; Wang, Guofang; Zhang, Yongbing
Abstract. In this paper, we introduce the Sasaki–Ricci flow to study the existence of η-Einstein metrics. In the positive case any η-Einstein metric can be homothetically transformed to a Sasaki–Einstein metric. Hence it is an odd-dimensional counterpart of the Kähler–Ricci flow. We prove its well-posedness and long-time existence. In the negative or null case the flow converges to the unique η-Einstein metric. In the positive case the convergence remains in general open. The paper can be viewed as an odd-dimensional counterpart of Cao’s results on the Kähler–Ricci flow.
Journal. Inter. J. of Math.; 2010; Vol 21, no. 7; 951-969.
Preprint preprint  Article article
24 Decomposition and minimality of Lagrangian submanifolds in nearly Kähler manifolds.
Schäfer, Lars; Smoczyk, Knut
Abstract. We show that Lagrangian submanifolds in six-dimensional nearly Kähler (non- Kähler) manifolds and in twistor spaces Z over quaternionic Kähler manifolds Q are minimal. Moreover, we prove that any Lagrangian submanifold L in a nearly Kähler manifold M splits into a product of two Lagrangian submanifolds for which one factor is Lagrangian in the strict nearly Kähler part of M and the other factor is Lagrangian in the Kähler part of M. Using this splitting theorem, we then describe Lagrangian submanifolds in nearly Kähler manifolds of dimensions six, eight, and ten.
Journal. Annals of Global Analysis and Geometry; 2010; Vol 37, no. 3; 221--240.
Preprint preprint  Article article
23 The hyperbolic mean curvature flow.
LeFloch, Phillipe; Smoczyk, Knut
Abstract. 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 from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.
Journal. Journal de Mathématiques Pures et Appliquées; 2008; Vol (9) 90, no. 6; 591--614.
Preprint preprint  Article article
22 Mean curvature flow of monotone Lagrangian submanifolds.
Groh, Konrad; Schwarz, Matthias; Smoczyk, Knut; Zehmisch, Kai
Abstract. We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in Cn.
Journal. Mathematische Zeitschrift; 2007; Vol 257, no. 2; 295--327.
Preprint preprint  Article article
21 Bernstein type theorems with flat normal bundle.
Smoczyk, Knut; Wang, Guofang; Xin, Yuanlong
Abstract. We prove Bernstein type theorems for minimal n-submanifolds in Rn+p with flat normal bundle. Those are natural generalizations of the corresponding results of Ecker-Huisken and Schoen-Simon-Yau for minimal hypersurfaces.
Journal. Calculus of Variations; 2006; Vol. 26; 57--67.
Preprint preprint  Article article
20  A representation formula for the inverse harmonic mean curvature flow.
Smoczyk, Knut
Abstract. Consider a smooth family of embedded, strictly convex hypersurfaces in Rn+1 evolving by the inverse harmonic mean curvature flow d/dt F = H-1ν. Surprisingly, we can determine the explicit solution of this nonlinear parabolic equation with some Fourier analysis. More precisely, there exists a representation formula for the evolving hypersurfaces that can be expressed in terms of the heat kernel on Sn and the initial support function.
Journal. Elemente der Mathematik; 2005; Vol. 60; 57--65.
Preprint preprint  Article article
19 Self-shrinkers of the mean curvature flow in arbitrary codimension.
Smoczyk, Knut
Abstract. In this paper we study self-similar solutions Mm in Rn of the mean curvature flow in arbitrary codimension. Self-similar curves in R2 have been completely classified by Abresch & Langer and this result can be applied to curves in Rn as well. A submanifold Mm in Rn is called spherical, if it is contained in a sphere. Obviously, spherical self-shrinkers of the mean curvature flow coincide with minimal submanifolds of the sphere. For hypersurfaces Mm in Rn+1, m>1, Huisken showed that compact self-shrinkers with positive scalar mean curvature are spheres. We prove the following extension: A compact self-similar solution Mm in Rn, m>1, is spherical, if and only if the mean curvature vector H is non-vanishing and the principal normal is parallel in the normal bundle. We also give a classification of complete noncompact self-shrinkers of that type.
Journal. IMRN; 2005; Vol 48; 2983--3004.
Preprint preprint  Article article
18 Longtime existence of the Lagrangian mean curvature flow.
Smoczyk, Knut
Abstract. Given a compact Lagrangian submanifold in flat space evolving by its mean curvature, we prove uniform C2,α-bounds in space and C2-estimates in time for the underlying Monge-Ampére equation under weak and natural assumptions on the initial Lagrangian submanifold. This implies longtime existence and convergence of the Lagrangian mean curvature flow. In the 2-dimensional case we can relax our assumptions and obtain two independent proofs for the same result.
Journal. Calculus of Variations; 2004; Vol. 20; 25--46.
Preprint preprint  Article article
17 Neumann and second boundary value problems for Hessian and Gauss curvature flows.
Schnürer, Oliver; Smoczyk, Knut
Abstract. We consider the flow of a strictly convex hypersurface driven by the Gauss curvature. For the Neumann boundary value problem and for the second boundary value problem we show that such a flow exists for all times and converges eventually to a solution of the prescribed Gauss curvature equation. We also discuss oblique boundary value problems and flows for Hessian equations.
Journal. Annales de l'Institut Henri Poincare (C) Non Linear Analysis; 2003; Vol. 20; 1043--1073.
Preprint preprint  Article article
16 Closed Legendre geodesics in Sasaki manifolds.
Smoczyk, Knut
Abstract. If L is a Legendre submanifold in a Sasaki manifold M, then the mean curvature flow does not preserve the Legendre condition. We define a kind of mean curvature flow for Legendre submanifolds which slightly differs from the standard one and then we prove that closed Legendre curves L in a Sasaki space form M converge to closed Legendre geodesics, if k²+σ+3≤0 and rot(L)=0, where σ denotes the sectional curvature of the contact plane ξ and k and rot(L) are the curvature respectively the rotation number of L. If rot(L)=0, we obtain convergence of a subsequence to Legendre curves with constant curvature. In case σ+3≤0 and if the Legendre angle α of the initial curve satisfies osc (α)≤π, then we also prove convergence to a closed Legendre geodesic.
Journal. New York Journal of Mathematics; 2003; Vol. 9; 23--47.
Preprint preprint  Article article
15 Mean curvature flows for Lagrangian submanifolds with convex potentials.
Smoczyk, Knut; Wang, Mu-Tao
Abstract. 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 of a Lagrangian graph in T2n is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold. We also discuss various conditions on the potential function that guarantee global existence and convergence.
Journal. Journal of Differential Geometry; 2002; Vol. 62; 243--257.
Preprint preprint  Article article
14 Prescribing the Maslov form of Lagrangian immersions.
Smoczyk, Knut
Abstract. We formulate and apply a modified Lagrangian mean curvature flow to prescribe the Maslov form of Lagrangian immersions in Cn. We prove longtime existence results and derive optimal results for curves.
Journal. Geometriae Dedicata; 2002; Vol. 91; 59--69.
Preprint preprint  Article article
13 Angle theorems for the Lagrangian mean curvature flow.
Smoczyk, Knut
Abstract. We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle α for the corresponding Lagrangian submanifold in the cross product space L × M satisfies osc(α)≤π. If one considers a 4-dimensional Kähler-Einstein manifold M of nonpositive scalar curvature that admits two complex structures J,K which commute and assumes that L⊂M is a compact oriented Lagrangian submanifold w.r.t. J such that the Kähler form ω w.r.t. K restricted to L is positive and osc(α)≤π, then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. ω.
Journal. Mathematische Zeitschrift; 2002; Vol. 240; 849--883.
Preprint preprint  Article article
12 Evolution of hypersurfaces in central force fields.
Schnürer, Oliver; Smoczyk, Knut
Abstract. We consider flows of hypersurfaces in Rn+1 decreasing the energy induced by radially symmetric potentials. These flows are similar to the mean curvature flow but different phenomena occur. We show for a natural class of potentials that hypersurfaces converge smoothly to a uniquely determined sphere if they satisfy a strengthened starshapedness condition at the beginning.
Journal. Journal für die reine und angewandte Mathematik (Crelle's Journal); 2002; Vol. 550; 77--95.
Preprint preprint  Article article
11 Note on the spectrum of the Hodge-Laplacian for k-forms on minimal Legendre submanifolds in S2n+1
Smoczyk, Knut
Abstract. Given a minimal Legendre immersion L in S2n+1 and n≥k≥1 we prove that n+1-k is an eigenvalue of the Hodge-Laplacian acting on k and (k-1)-forms on L. In particular we show that the eigenspaces Eigk(n+1-k) and Eigk-1(n+1-k) are at least of dimension n over k.
Journal. Calculus of Variations; 2002; Vol. 14; 107-113.
Preprint preprint  Article article
10 A relation between mean curvature flow solitons and minimal submanifolds.
Smoczyk, Knut
Abstract. We derive a one to one correspondence between conformal solitons of the mean curvature flow in an ambient space N and minimal submanifolds in a different ambient space N, where N equals RxN equipped with a warped product metric and show that a submanifold in N converges to a conformal soliton under the mean curvature flow in N if and only if its associated submanifold in N converges to a minimal submanifold under a rescaled mean curvature flow in N. We then define a notion of stability for conformal solitons and obtain Lp-estimates as well as pointwise estimates for the curvature of stable solitons.
Journal. Mathematischen Nachrichten; 2001; Vol. 229; 175--186.
Preprint preprint  Article article
09 Soliton solutions for the mean curvature flow.
Hungerbühler, Norbert; Smoczyk, Knut
Abstract. We consider soliton solutions of the mean curvature flow, i.e., solutions which move under the mean curvature flow by a group of isometries of the ambient manifold. Several examples of solitons on manifolds are discussed. Moreover we present a local existence result for rotating solitons. We also prove global existence and stability for perturbed initial data close to a local soliton.
Journal. Differential Integral Equations; 2000; Vol. 13; 1321--1345.
Preprint preprint  Article article
08 Remarks on the inverse mean curvature flow.
Smoczyk, Knut
Abstract. In this short note we investigate the regularity of 2-surfaces evolving by its inverse mean curvature in an asymptotically flat Riemannian 3-manifold and derive an a-priori bound for the second fundamental form in terms of a quantity depending on the mean curvature and the elapsed time. This partially solves one of the 5 questions posed in a paper by Huisken and Ilmanen. The proof relies on the special geometry of asymptotically flat Riemannian manifolds and on the fact that the dimension of the evolving surface is 2.
Journal. Asian J. Math.; 2000; Vol. 4; 331--336.
Preprint preprint  Article article
07 The Lagrangian mean curvature flow (Der Lagrangesche mittlere Krümmungsfluss).
Smoczyk, Knut
Abstract. In my Habilitation thesis I give an introduction to and develop many concepts in the Lagrangian mean curvature flow. It is shown that no closed self-similarly shrinking Lagrangian solutions exist, if the Maslov class is trivial. Moreover, if the eigenvalues of the generating function of a Lagrangian graph in R2n are bounded between -1 and 1, then the Lagrangian mean curvature flow preserves this condition.
Journal. Universität Leipzig (Habilitation), 2000.
Preprint preprint
06 Nonexistence of minimal Lagrangian spheres in hyper-Kähler manifolds.
Smoczyk, Knut
Abstract. We prove that for n>1 one cannot immerse S2n as a minimal Lagrangian manifold into a hyper-Kähler manifold. More generally we show that any minimal Lagrangian immersion of an orientable closed manifold L2n into a hyper-Kähler manifold H4n must have nonvanishing second Betti number and that in case the second Betti number is 1, L2n is a Kähler manifold and more precisely a Kähler submanifold in H4n w.r.t. one of the complex structures on H4n. In addition we derive a result for the other Betti numbers.
Journal. Calculus of Variations; 2000; Vol. 10; 41--48.
Preprint preprint  Article article
05 Harnack inequality for the Lagrangian mean curvature flow.
Smoczyk, Knut
Abstract. If Ln is a Lagrangian manifold immersed into a Kähler-Einstein manifold, nothing is known about its behavior under the mean curvature flow. As a first result we derive a Harnack inequality for the mean curvature potential of compact Lagrangian immersions Ln immersed into R2n .
Journal. Calculus of Variations; 1999; Vol. 8; 247--258.
Preprint preprint  Article article
04 Starshaped hypersurfaces and the mean curvature flow.
Smoczyk, Knut
Abstract. Under the assumption of two a-priori bounds for the mean curvature, we are able to generalize a recent result due to Huisken and Sinestrari, valid for mean convex surfaces, to a much larger class. In particular we will demonstrate that these a-priori bounds are satis ed for a class of surfaces including meanconvex as well as starshaped surfaces and a variety of manifolds that are close to them. This gives a classification of the possible singularities for these surfaces in the case n=2. In addition we prove that under certain initial conditions some of them become mean convex before the first singularity occurs.
Journal. manuscripta mathematica; 1998; Vol. 95; 225--236.
Preprint preprint  Article article
03 Harnack inequalities for curvature flows depending on mean curvature.
Smoczyk, Knut
Abstract. We prove Harnack inequalities for parabolic flows of compact orientable hypersurfaces in Rn+1, where the normal velocity is given by a smooth function f depending only on the mean curvature. We use these estimates to prove longtime existence of solutions in some highly nonlinear cases. In addition we prove that compact selfsimilar solutions with constant mean curvature must be spheres and that compact selfsimilar solutions with nonconstant mean curvature can only occur in the case, where f(x)=Aαxα with two constants A,α.
Journal. New York Journal of Mathematics; 1997; Vol. 3; 103--118.
Preprint preprint  Article article
02 Symmetric hypersurfaces in Riemannian manifolds contracting to Lie-groups by their mean curvature.
Smoczyk, Knut
Abstract. This paper concerns the deformation by mean curvature of hypersurfaces M in Riemannian spaces N that are invariant under a subgroup of the isometry-group on N. We show that the hypersurfaces contract to this subgroup, if the cross-section satis es a strong convexity assumption.
Journal. Calculus of Variations; 1996; Vol. 4; 155--170.
Preprint preprint  Article article
01 The symmetric doughnut evolving by its mean curvature.
Smoczyk, Knut
Abstract. In this paper we discuss the behavior of rotationally symmetric hypersurfaces S1xM in Rn+1 and we prove that the hypersurfaces contract to S1, if the cross section M satisfies a natural convexity assumption.
Journal. Hokkaido Mathematical Journal; 1994; Vol. 23; 523--547.
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Letzte Änderung: 03.02.2014