Horo-Convex Hypersurfaces with Prescribed Shifted Gauss Curvatures in $$\pmb {\mathbb {H}}^{n+1}$$ H n + 1
- PDF / 326,459 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 52 Downloads / 168 Views
Horo-Convex Hypersurfaces with Prescribed Shifted Gauss Curvatures in H n+1 Li Chen1 · Qiang Tu1
· Kang Xiao1
Received: 3 May 2020 / Accepted: 3 October 2020 © Mathematica Josephina, Inc. 2020
Abstract In this paper, we consider prescribed shifted Gauss curvatures equations for horoconvex hypersurfaces in Hn+1 . Under some sufficient conditions, we obtain an existence result by the standard degree theory based on the a priori estimates for solutions to the equations. Different from the prescribed Weingarten curvatures problem in space forms, we do not impose a sign condition for the radial derivative of the functions in the right-hand side of the equations to prove the existence due to the horo-covexity of hypersurfaces in Hn+1 . Keywords Shifted Gauss curvature · Horo-convex · Mong–Ampére type equation Mathematics Subject Classification Primary 35J96, 52A39 · Secondary 53A05
1 Introduction Different from hypersurfaces in Rn+1 , there are four different kinds of convexity for hypersurfaces in Hn+1 [1,2]. One of them is the horo-convexity which is defined as follows.
This research was supported by funds from Hubei Provincial Department of Education Key Projects D20181003.
B
Qiang Tu [email protected] Li Chen [email protected] Kang Xiao [email protected]
1
Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, People’s Republic of China
123
L. Chen et al.
Definition 1.1 A smooth oriented hypersurface M ⊂ Hn+1 is called horo-convex if, for a given orientation, κi ( p) > 1 for all p ∈ M and 1 ≤ i ≤ n, where κ = (κ1 , ..., κn ) are the principal curvatures of M ⊂ Hn+1 which are defined by eigenvalues of the j Weingarten matrix W = (h i ). Geometrically, a hypersurface M ⊂ Hn+1 is called horo-convex if and only if it is convex by horospheres, where the horospheres in hyperbolic space are hypersurfaces with constant principal curvatures equal to 1 everywhere for its inward orientation, the one that points into the horoball bounded by the horosphere. In [18], the authors suggest that horospheres can be naturally regarded in many ways as hyperplanes in the hyperbolic space Hn+1 . This fact implies the similarity between the horo-convixty in Hn+1 and the convexity in Rn+1 from geometric aspect. Furthermore, this interesting formal similarities, or the general similarities between the geomerty of horo-convex regions in Hn+1 (that is, regions which are given by the intersection of a collection of horo-balls) and that of convex Euclidean bodies have been deeply explored in [10,18]. In particular, Andrews-Chen-Wei [10] introduce the shifted Weingarten matrix W−I for the hypersurface M ⊂ Hn+1 along the line that the convexity of hypersurfaces in Rn+1 can be describes by the positive definite of their Weingarten matrix. Clearly, M ⊂ Hn+1 is horo-convex if and only if the shifted Weingarten matrix W − I is positive definite. Moreover, they [10] define the shifted principal curvatures by (λ1 , ..., λn ) := (κ1 − 1, ..., κn − 1), which are eigenvalues of
Data Loading...
