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October 8, 2025Open Access

Asymptotic Plateau problem for 2-convex hypersurface in H⁴

Key Points

The main finding shows the existence of smooth complete 2-convex hypersurfaces with specific curvature conditions.
The key evidence proves a critical curvature equation involving three principal curvatures matches a constant's cubic relation.
An observational analysis explores geometry within hyperbolic space, addressing prescribed boundary conditions at infinity.
This research may enable further exploration in geometric analysis, potentially influencing higher-dimensional geometric theories.

Abstract

We prove the existence of smooth complete 2-convex hypersurface which satisfies prescribed curvature equation (₁ + ₂) (₁ + ₃) (₂ + ₃) = (2) ³ and has prescribed asymptotic boundary at infinity of hyperbolic space of dimension 4, where (0, 1) is a constant. We also prove the existence for ₖ (₂ + + ₙ, , ₁ + + ₍ - ₁) = Cₙᵏ (n - 1) ᵏ ᵏ with k < n in H^n + 1.

Asymptotic Plateau problem for 2-convex hypersurface in H⁴

Key Points

Abstract

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