Convexity of 𝜆-hypersurfaces

Lee, Tang-Kai

doi:10.1090/proc/15707

Convexity of $\lambda$-hypersurfaces
HTML articles powered by AMS MathViewer

by Tang-Kai Lee

Proc. Amer. Math. Soc. 150 (2022), 1735-1744

DOI: https://doi.org/10.1090/proc/15707

Published electronically: January 26, 2022

PDF | Request permission

Abstract:

We prove that any $n$-dimensional closed mean convex $\lambda$- hypersurface is convex if $\lambda \le 0.$ This generalizes Guang’s work on $2$-dimensional strictly mean convex $\lambda$-hypersurfaces. As a corollary, we obtain a gap theorem for closed $\lambda$-hypersurfaces with $\lambda \le 0.$

References

U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), no. 2, 175–196. MR 845704, DOI 10.4310/jdg/1214440025
F. Barthe, An isoperimetric result for the Gaussian measure and unconditional sets, Bull. London Math. Soc. 33 (2001), no. 4, 408–416. MR 1832552, DOI 10.1017/S0024609301008141
Christer Borell, The Ehrhard inequality, C. R. Math. Acad. Sci. Paris 337 (2003), no. 10, 663–666 (English, with English and French summaries). MR 2030108, DOI 10.1016/j.crma.2003.09.031
Kenneth A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978. MR 0485012
Jui-En Chang, 1-dimensional solutions of the $\lambda$-self shrinkers, Geom. Dedicata 189 (2017), 97–112. MR 3667341, DOI 10.1007/s10711-017-0219-z
Qing-Ming Cheng and Guoxin Wei, Complete $\lambda$-hypersurfaces of weighted volume-preserving mean curvature flow, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 32, 21. MR 3763110, DOI 10.1007/s00526-018-1303-4
Tobias Holck Colding and William P. Minicozzi II, A course in minimal surfaces, Graduate Studies in Mathematics, vol. 121, American Mathematical Society, Providence, RI, 2011. MR 2780140, DOI 10.1090/gsm/121
Tobias H. Colding and William P. Minicozzi II, Generic mean curvature flow I: generic singularities, Ann. of Math. (2) 175 (2012), no. 2, 755–833. MR 2993752, DOI 10.4007/annals.2012.175.2.7
Tobias Holck Colding and William P. Minicozzi II, Uniqueness of blowups and Łojasiewicz inequalities, Ann. of Math. (2) 182 (2015), no. 1, 221–285. MR 3374960, DOI 10.4007/annals.2015.182.1.5
Qiang Guang, Gap and rigidity theorems of $\lambda$-hypersurfaces, Proc. Amer. Math. Soc. 146 (2018), no. 10, 4459–4471. MR 3834671, DOI 10.1090/proc/14111
Qiang Guang, A note on mean convex $\lambda$-surfaces in $\Bbb R^3$, Proc. Amer. Math. Soc. 149 (2021), no. 3, 1259–1266. MR 4211879, DOI 10.1090/proc/15297
Steven Heilman, Symmetric convex sets with minimal Gaussian surface area, Amer. J. Math. 143 (2021), no. 1, 53–94. MR 4201779, DOI 10.1353/ajm.2021.0000
D. Hoffman, T. Ilmanen, F. Martín, and B. White, Notes on translating solitons for mean curvature flow, 2019. arXiv:1901.09101.
Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299. MR 1030675
Gerhard Huisken, Local and global behaviour of hypersurfaces moving by mean curvature, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 175–191. MR 1216584, DOI 10.1090/pspum/054.1/1216584
T. Ilmanen, Singularities of mean curvature flow of surfaces. 1995, preprint on webpage at http://www.math.ethz.ch/~ilmanen/papers/pub.html.
Michel Ledoux, Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space, Bull. Sci. Math. 118 (1994), no. 6, 485–510. MR 1309086
Matthew McGonagle and John Ross, The hyperplane is the only stable, smooth solution to the isoperimetric problem in Gaussian space, Geom. Dedicata 178 (2015), 277–296. MR 3397495, DOI 10.1007/s10711-015-0057-9
Donald H. Singley, Smoothness theorems for the principal curvatures and principal vectors of a hypersurface, Rocky Mountain J. Math. 5 (1975), 135–144. MR 367878, DOI 10.1216/RMJ-1975-5-1-135
Joel Spruck and Ling Xiao, Complete translating solitons to the mean curvature flow in $\Bbb R^3$ with nonnegative mean curvature, Amer. J. Math. 142 (2020), no. 3, 993–1015. MR 4101337, DOI 10.1353/ajm.2020.0023
Brian White, Partial regularity of mean-convex hypersurfaces flowing by mean curvature, Internat. Math. Res. Notices 4 (1994), 186 ff., approx. 8 pp.}, issn=1073-7928, review= MR 1266114, doi=10.1155/S1073792894000206, DOI 10.1155/S1073792894000206