Translating solutions to Lagrangian mean curvature flow

Authors:
André Neves and Gang Tian

Journal:
Trans. Amer. Math. Soc. **365** (2013), 5655-5680

MSC (2010):
Primary 53C44, 53D12

DOI:
https://doi.org/10.1090/S0002-9947-2013-05649-8

Published electronically:
June 26, 2013

MathSciNet review:
3091260

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an bound on the mean curvature are planes and that almost-calibrated translating solutions which are static are also planes. Recent work of D. Joyce, Y.-I. Lee, and M.-P. Tsui shows that these conditions are optimal.

**1.**G. Huisken, Asymptotic behavior for singularities of the mean curvature flow,**J. Differential Geom. 31**(1990), 285-299. MR**1030675 (90m:53016)****2.**T. Ilmanen, Elliptic Regularization and Partial Regularity for Motion by Mean Curvature.**Mem. Amer. Math. Soc. 108**(1994), 1994. MR**1196160 (95d:49060)****3.**T. Ilmanen, Singularities of Mean Curvature Flow of Surfaces. Preprint.**4.**D. Joyce, Y.-I. Lee, M.-P. Tsui, Self-similar solutions and translating solitons for Lagrangian mean curvature flow,**J. Differential Geom. 84**(2010), no. 1, 127-161. MR**2629511 (2011f:53151)****5.**A. Neves, Singularities of Lagrangian Mean Curvature Flow: Zero-Maslov class case.**Invent. Math. 168**(2007), 449-484. MR**2299559 (2008d:53092)****6.**G. Perelman, The entropy formula for the Ricci flow and its geometric applications. preprint.**7.**L. Simon, Lectures on geometric measure theory,**Proceedings of the Centre for Mathematical Analysis, Australian National University, 3**. MR**756417 (87a:49001)****8.**K. Smoczyk, Existence of solitons for the Lagrangian mean curvature flow. ETH Zürich; 1998; preprint.**9.**B. White, The size of the singular set in mean curvature flow of mean-convex sets.**J. Amer. Math. Soc. 13**(2000), 665-695. MR**1758759 (2001j:53098)****10.**B. White, The nature of singularities in mean curvature flow of mean-convex sets.**J. Amer. Math. Soc. 16**(2003), 123-138 . MR**1937202 (2003g:53121)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
53C44,
53D12

Retrieve articles in all journals with MSC (2010): 53C44, 53D12

Additional Information

**André Neves**

Affiliation:
Department of Pure Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, United Kingdom

Email:
aneves@imperial.ac.uk

**Gang Tian**

Affiliation:
Department of Mathematics, Fine Hall, Princeton University, Princeton, New Jersey 08544

Email:
tian@math.princeton.edu

DOI:
https://doi.org/10.1090/S0002-9947-2013-05649-8

Received by editor(s):
June 9, 2011

Published electronically:
June 26, 2013

Additional Notes:
The author was partially supported by NSF grant DMS-06-04164.

Article copyright:
© Copyright 2013
American Mathematical Society