Optimal global regularity for minimal graphs over convex domains in hyperbolic space

Abstract

We use the concept of the inside-(a, η, h) domain to construct a subsolution to the Dirichlet problem for minimal graphs over convex domains in hyperbolic space. As an application, we prove that the Hölder exponent \max \{1/a,1/(n+1)\} for the problem is optimal for any a\in [2,+\infty ].

Keywords

Minimal graph equation / optimal regularity / global regularity

References

[1] Anderson A. Complete minimal varieties in hyperbolic space. Invent Math, 1982, 67: 477–494

[2] Anderson A. Complete minimal hypersurfaces in hyperbolic n-manifolds. Comment Math Helv, 1983, 58: 264–290

[3] Han Q, Shen W, Wang Y. Optimal regularity of minimal graphs in the hyperbolic space. Calc Var Partial Differential Equations, 2016, 55: 1–19

[4] Hardt R, Lin F H. Regularity at infinity for area-minimizing hypersurfaces in hyperbolic space. Invent Math, 1987, 88: 217–224

[5] Jian H Y, Li Y. Optimal boundary regularity for a singular Monge-Ampère equation. J Differential Equations, 2018, 264: 6873–6890

[6] Jian H Y, Li Y. Global regularity for minimal graphs over convex domains in hyperbolic space. J Differential Equations, 2021, 271: 963–978

[7] Lin F H. On the Dirichlet problem for minimal graphs in the hyperbolic space. Invent Math, 1989, 96: 593–612

[8] Lin F H. Asymtotic behavior of area-minimizing currents in hyperbolic space. Comm Pure Appl Math, 1989, 42: 229–242

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