MEAN CURVATURE FLOW
Course bibliography (2013-2014)- Alex Freire
A. BOOKS
K. Ecker, Regularity Theory for MCF, Birkhauser 2004
3. Local estimates via maximum
principle/ 4. Integral estimates and monotonicity/ 5. Regularity
at the first singular time
C. Mantegazza, Lectures on MCF, Birkhauser 2011
2. Evolution of geometric quantities/
3. Monotonicity and type I singularities/ 4. Type II
singularities
Survey: B. White Evolution of
curves and surfaces by mean curvature, ICM 2002
B. RESEARCH PAPERS
1.1 Classic (differential geometry)
G. Huisken Flow by MC of convex
surfaces into spheres, JDG 1984
K. Ecker, G.Huisken Mean curvature evolution of entire graphs, Annals
1989
K. Ecker, G. Huisken Interior estimates for hypersurfaces moving
by MC, Inventiones 1991
G. Huisken, Asymptotic behavior of
singularities for MCF, JDG 1990
B. Andrews, Contraction of convex
hypersurfaces in Euclidean space, CV&PDE 1994
R. Hamilton, Harnack estimate for MCF, JDG 1995
J. Velazquez, Curvature Blow-up in perturbations of minimal cones
evolving by MCF, Annali Pisa 1994
S. Angenent, J. Velazquez, Degenerate neck-pinches in MCF, J. Reine u.
Angewandte Math 1997
1.2 Classic (weak solutions)
L.C. Evans, J. Spruck, Motion of Level
Sets by MC (I), JDG 1991
YG Chen, Y.Giga, S.Goto, Uniqueness & Existence of Viscosity
solutions for generalized MCF, JDG 1991
LC Evans, HM Soner, PE Souganidis Phase Transitions and generalized
MCF, CPAM 1992
T. Ilmanen, Generalized flow of sets by MC on a manifold, Indiana U
Math. J 1992
HM Soner, Motion of a set by the curvature of its boundary, JDE 1993
T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion
by MC, JDG 1993
T. Ilmanen, Elliptic regularization and partial regularity for motion
by MC, Memoirs AMS 1993
S. Altschuler, S. Angenent, Y. Giga,
MCF through singularities for surfaces of Revolution, J. Geom. Analysis
1995
2.1 Semi-modern
G. Huisken, C. Sinestrari, MCF
singularities for mean-convex surfaces, CV&PDE 1999
K. Smoczyk, Starshaped hypersurfaces and MCF, Manuscripta 1998
G. Huisken, C. Sinestrari, Convexity
estimates for MCF and singularities of mean-convex surfaces, Acta Math.
1999
B. White, The size of the singular set in MCF of mean-convex
sets, JAMS 2000
B. White, The nature of singularities in MCF of mean-convex sets, JAMS
2003
B White, A local regularity theorem for MCF,
Annals 2005
2.2 Inverse mean curvature flow
J. Urbas, Expansion of starshaped
hypersurfaces by symmetric functions of principal curvatures, Math Z.
1990
C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, JDG
1990
B. Chow, R. Gulliver, Aleksandrov reflection and geometric
evolution of hypersurfaces, CA&G 2001
P. Guan, J. Li, Quermassintegral inequalities for starshaped
domains, Arxiv (2007)
G. Huisken, T. Ilmanen,
The IMCF and the Riemannian Penrose inequality, JDG 2001
G. Huisken, T. Ilmanen, Higher
Regularity of the IMCF, JDG 2008
2.3 Recent
G. Huisken, C. Sinestrari, MCF with
surgeries for 2-convex hypersurfaces, Inventiones 2009
B. Andrews, Non-collapsing in mean-convex MCF, Arxiv 2011
XJ Wang, Convex solutions to the MCF, Annals 2011
B. White, Subsequent singularities in MCF, ArXiv 2011, 2013
TH Colding, WP Minicozzi, Generic
MCF(I): generic singularities, Annals 2012
R. Haslhofer, B. Kleiner, MCF of
mean-convex hypersurfaces, ArXiv (2013)
S. Brendle, Sharp bound for the
inscribed radius under MCF, ArXiv (2013)
S. Brendle, G. Huisken, MCF with
surgery of mean-convex surfaces in R3, ArXiv (2013)
TH Colding, WP Minicozzi, Rigidity of generic singularities of
MCF, ArXiv (2013)
TH Colding, WP Minicozzi, Uniqueness of blowups and Lojasiewicz
inequalities, ArXiv (2013)
B. Andrews, M. Langford, Two-sided non-collapsing curvature flows,
ArXiv 2013/2014