Translating Solutions of the Nonparametric Mean Curvature Flow with Nonzero Neumann Boundary Data in Product Manifold M ⁿ × ℝ

Abstract

In this paper, the authors can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold M ⁿ × ℝ, where M ⁿ is an n-dimensional (n ≥ 2) complete Riemannian manifold with nonnegative Ricci curvature, and ℝ is the Euclidean 1-space.

Keywords

Translating solutions / Singularity / Nonparametric mean curvature flow / Convexity / Ricci curvature

References

[1] Abresch U, Rosenberg H. A Hopf differential for constant mean curvature surfaces in ${\mathbb{S}^2} \times \mathbb{R}$ and ℍ² × ℍ. Acta Math., 2004, 193: 141-174

[2] Altschuler S-J, Wu L-F. Convergence to translating solutions for a class of quasilinear parabolic boundary problems. Math. Ann., 1993, 295: 761-765

[3] Altschuler S-J, Wu L-F. Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var. Partial Differential Equations, 1994, 2: 101-111

[4] Angenent S-B, Velázquez J-J-L. Asymptotic shape of cusp singularities in curve shortening. Duke Math. J., 1995, 77: 71-110

[5] Angenent S-B, Velázquez J-J-L. Degenerate neckpinches in mean curvature flow. J. Reine Angew. Math., 1997, 482: 15-66

[6] Bandle C. Isoperimetric Inequalities and Applications, 1980, Boston: Pitman

[7] Chen L, Hu D-D, Mao J, Xiang N. Translating surfaces of the nonparametric mean curvature flow in Lorentz manifold M ² × ℝ. Chinese Ann. Math. Ser. B, 2021, 42(2): 297-310

[8] Gao, Y. and Mao, J., Translating solitons to the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold M ⁿ × ℝ, II (in preprint).

[9] Gao Y, Mao J, Song C-L. Existence and uniqueness of solutions to the constant mean curvature equation with nonzero Neumann boundary data in product manifold M ⁿ × ℝ. Acta Math. Sci. Ser. A, 2020, 40(6): 1525-1536 (in Chinese)

[10] Guan, B., Mean curvature motion of nonparametric hypersurfaces with contact angle condition, Elliptic and parabolic methods in geoemtry (Minneapolis, MN, 1994), 1996, 47–56.

[11] Huisken G. Flow by mean curvature of convex surfaces into spheres. J. Differential Geom., 1984, 20: 237-266

[12] Huisken G. Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math., 1986, 84: 463-480

[13] Huisken G. Non-parametric mean curvature evolution with boundary conditions. J. Differential Equat., 1989, 77: 369-378

[14] Jenkins H, Serrin J. The Dirichlet problem for the minimal surface equation in higher dimensions. J. Reine Angew. Math., 1968, 229: 170-187

[15] Ma X-N, Wang P-H, Wei W. Constant mean curvature surfaces and mean curvature flow with nonzero Neumann boundary conditions on strictly convex domains. J. Funct. Anal., 2018, 274: 252-277

[16] Mao J. A new way to Dirichlet problems for minimal surface system in arbitrary dimensions and codimensions. Kyushu J. Math., 2015, 69(1): 1-9

[17] Mazet L, Rodríguez M-M, Rosenberg H. Periodic constant mean curvature surfaces in ℍ² × ℝ. Asian J. Math., 2014, 18: 829-858

[18] Meeks W-H, Rosenberg H. Stable minimal surfaces in M × ℝ. J. Differential Geom., 2004, 68: 515-534

[19] Osserman R. The isoperimetric inequality. Bull. Amer. Math. Soc., 1978, 84: 1182-1238

[20] Rosenberg H, Schulze F, Spruck J. The half-space property and entire positive minimal graphs in M × ℝ. J. Differential Geom., 2013, 95: 321-336

[21] Topping P. Mean curvature flow and geometric inequalities. J. Reine Angew. Math., 1998, 503: 47-61

[22] Wang J, Wei W, Xu J-J. Translating solutions of non-parametric mean curvature flows with capillary-type boundary value problems. Commun. Pure Appl. Anal., 2019, 18(6): 3243-3265

[23] Wang M-T. Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension. Invent. Math., 2002, 148(3): 525-543

[24] Wang M-T. The Dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions. Commun. Pure. Appl. Math., 2004, 57(2): 267-281

[25] Zhou H-Y. Nonparametric mean curvature type flows of graphs with contact angle conditions. Int. Math. Res. Not., 2018, 19: 6026-6069