




36 
Curvature decay estimates for mean curvature flow in higher codimensions.
Smoczyk, Knut; Tsui, MaoPei; Wang, MuTao
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

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35 
Evolution of contractions by mean curvature flow.
SavasHalilaj, 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.

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34 
Homotopy of area decreasing maps by mean curvature flow.
SavasHalilaj, 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; 455473.

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Article 

33 
Bernstein theorems for length and area decreasing minimal maps.
SavasHalilaj, 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).

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Article 

32 
Evolution of spacelike surfaces in antiDe Sitter space by their Lagrangian angle.
Smoczyk, Knut
Abstract.
We study spacelike hypersurfaces M in an antiDe
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
spacelike surface, if the Gauss curvature K of the initial surface M satisfies k<K<k.
Journal.
Mathematische Annalen; 2013, Vol 355; no. 4; 14431468.

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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, 231274.

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Article 

30 
Generalized Lagrangian mean curvature flows in symplectic manifolds.
Smoczyk, Knut; Wang, MuTao
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 KählerEinstein manifolds where the connection
is the LeviCivita 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; 129140.

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29 
Mean curvature flow of spacelike Lagrangian submanifolds in almost paraKähler manifolds.
Chursin, Mykhaylo; Schäfer, Lars; Smoczyk, Knut
Abstract.
Given an almost paraKähler manifold equipped with
a metric and paracomplex connection, we define a generalized second
fundamental form and generalized mean curvature vector of
spacelike 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 paraKähler structure
is integrable, the
flow coincides with the classical mean curvature
flow in the pseudoRiemannian context.
Journal.
Calculus of Variations; 2011; Vol 41; no. 12; 111125.

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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 

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 selfshrinking
quadrics.
Journal.
Analysis (Munich); 2011; Vol. 31; 91102.

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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.
EastWest J. Math.; 2010, Special Vol., 292305.

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25 
The SasakiRicci flow.
Smoczyk, Knut; Wang, Guofang; Zhang, Yongbing
Abstract.
In this paper, we introduce the Sasaki–Ricci flow to study the existence of ηEinstein
metrics. In the positive case any ηEinstein metric can be homothetically transformed to
a Sasaki–Einstein metric. Hence it is an odddimensional counterpart of the Kähler–Ricci
flow. We prove its wellposedness and longtime existence. In the negative or null case
the flow converges to the unique ηEinstein metric. In the positive case the convergence
remains in general open. The paper can be viewed as an odddimensional counterpart
of Cao’s results on the Kähler–Ricci flow.
Journal.
Inter. J. of Math.; 2010; Vol 21, no. 7; 951969.

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Article 

24 
Decomposition and minimality of Lagrangian submanifolds in nearly Kähler manifolds.
Schäfer, Lars; Smoczyk, Knut
Abstract.
We show that Lagrangian submanifolds in sixdimensional 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; 221240.

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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 initialvalue problem is locally wellposed 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 blowup 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 onedimensional graphs, a globalintime existence result is established.
Journal.
Journal de Mathématiques Pures et Appliquées; 2008; Vol (9) 90, no. 6; 591614.

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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 C^{n}.
Journal.
Mathematische Zeitschrift; 2007; Vol 257, no. 2; 295327.

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Article 

21 
Bernstein type theorems with flat normal bundle.
Smoczyk, Knut; Wang, Guofang; Xin, Yuanlong
Abstract.
We prove Bernstein type theorems for minimal nsubmanifolds in R^{n+p}
with flat normal bundle. Those are natural generalizations of the corresponding
results of EckerHuisken and SchoenSimonYau for minimal hypersurfaces.
Journal.
Calculus of Variations; 2006; Vol. 26; 5767.

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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 R^{n+1} evolving by the inverse harmonic mean
curvature flow
d/dt F = H^{1}ν. 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 S^{n} and
the initial support function.
Journal.
Elemente der Mathematik; 2005; Vol. 60; 5765.

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Article 

19 
Selfshrinkers of the mean curvature flow in arbitrary codimension.
Smoczyk, Knut
Abstract.
In this paper we study selfsimilar solutions M^{m} in R^{n} of the
mean curvature flow in arbitrary codimension. Selfsimilar curves in R^{2}
have been completely classified by Abresch & Langer and this result
can be applied to curves in R^{n} as well. A submanifold M^{m} in R^{n} is
called spherical, if it is contained in a sphere. Obviously, spherical selfshrinkers
of the mean curvature flow coincide with minimal submanifolds of the sphere.
For hypersurfaces M^{m} in R^{n+1}, m>1, Huisken showed that compact
selfshrinkers with positive scalar mean curvature are spheres. We prove
the following extension: A compact selfsimilar solution M^{m} in R^{n}, m>1, is
spherical, if and only if the mean curvature vector H is nonvanishing and the
principal normal is parallel in the normal bundle. We also give a classification
of complete noncompact selfshrinkers of that type.
Journal.
IMRN; 2005; Vol 48; 29833004.

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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 C^{2,α}bounds in space and
C^{2}estimates in time for the underlying MongeAmpére 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 2dimensional case we can relax our assumptions
and obtain two independent proofs for the same result.
Journal.
Calculus of Variations; 2004; Vol. 20; 2546.

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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; 10431073.

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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; 2347.

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Article 

15 
Mean curvature flows for Lagrangian submanifolds with convex potentials.
Smoczyk, Knut; Wang, MuTao
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 T^{2n} 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; 243257.

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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 C^{n}. We prove longtime existence results and
derive optimal results for curves.
Journal.
Geometriae Dedicata; 2002; Vol. 91; 5969.

Preprint 
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 4dimensional KählerEinstein 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; 849883.

Preprint 
Article 

12 
Evolution of hypersurfaces in central force fields.
Schnürer, Oliver; Smoczyk, Knut
Abstract.
We consider flows of hypersurfaces in R^{n+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; 7795.

Preprint 
Article 

11 
Note on the spectrum
of the HodgeLaplacian for kforms on minimal Legendre submanifolds in S^{2n+1}
Smoczyk, Knut
Abstract.
Given a minimal Legendre immersion L in S^{2n+1} and n≥k≥1 we prove that
n+1k is an eigenvalue of the HodgeLaplacian acting on k and (k1)forms on L.
In particular we show that the eigenspaces Eig_{k}(n+1k) and Eig_{k1}(n+1k) are at
least of dimension n over k.
Journal.
Calculus of Variations; 2002; Vol. 14; 107113.

Preprint 
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 L^{p}estimates as well as pointwise
estimates for the curvature of stable solitons.
Journal.
Mathematischen Nachrichten; 2001; Vol. 229; 175186.

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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; 13211345.

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Article 

08 
Remarks on the inverse mean curvature flow.
Smoczyk, Knut
Abstract.
In this short note we investigate the regularity of 2surfaces evolving by its inverse
mean curvature in an asymptotically flat Riemannian 3manifold and derive an apriori 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; 331336.

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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 selfsimilarly shrinking Lagrangian solutions exist, if the Maslov class is trivial. Moreover,
if the eigenvalues of the generating function of a Lagrangian graph in R^{2n} are bounded between 1 and 1, then
the Lagrangian mean curvature flow preserves this condition.
Journal.
Universität Leipzig (Habilitation), 2000.

Preprint 


06 
Nonexistence of minimal Lagrangian spheres in hyperKähler manifolds.
Smoczyk, Knut
Abstract.
We prove that for n>1 one cannot immerse S^{2n} as a minimal
Lagrangian manifold into a hyperKähler manifold. More generally we show
that any minimal Lagrangian immersion of an orientable closed manifold
L^{2n} into a hyperKähler manifold H^{4n} must have nonvanishing second Betti
number and that in case the second Betti number is 1, L^{2n} is a Kähler manifold and more precisely
a Kähler submanifold in H^{4n} w.r.t. one of the complex structures on H^{4n}.
In addition we derive a result for the other Betti numbers.
Journal.
Calculus of Variations; 2000; Vol. 10; 4148.

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Article 

05 
Harnack inequality for the Lagrangian mean curvature flow.
Smoczyk, Knut
Abstract.
If L^{n} is a Lagrangian manifold immersed into a KählerEinstein 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 L^{n} immersed into R^{2n} .
Journal.
Calculus of Variations; 1999; Vol. 8; 247258.

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Article 

04 
Starshaped hypersurfaces and the mean curvature flow.
Smoczyk, Knut
Abstract.
Under the assumption of two apriori 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 apriori bounds are satised 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; 225236.

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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 R^{n+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)=Aαx^{α} with two constants A,α.
Journal.
New York Journal of Mathematics; 1997; Vol. 3; 103118.

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Article 

02 
Symmetric hypersurfaces in Riemannian manifolds contracting to Liegroups 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 isometrygroup on N. We show that the hypersurfaces
contract to this subgroup, if the crosssection satises a strong convexity
assumption.
Journal.
Calculus of Variations; 1996; Vol. 4; 155170.

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Article 

01 
The symmetric doughnut evolving by its mean curvature.
Smoczyk, Knut
Abstract.
In this paper we discuss the behavior of rotationally symmetric hypersurfaces S^{1}xM in R^{n+1} and
we prove that the hypersurfaces contract to S^{1}, if the cross section M satisfies a natural convexity
assumption.
Journal.
Hokkaido Mathematical Journal; 1994; Vol. 23; 523547.

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Letzte Änderung: 03.02.2014



