Convex hypersurfaces of prescribed curvature

By Bo Guan, Pengfei Guan. 2002

168 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/299b962302f908bfbbe4da90eac5e726b6cfcbb8

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Volume-preserving flow by powers of the mth mean curvature (2009, methodology): “ations ofthese quantities. Various problemsinvolving these curvatures have been considered in the literature on geometric analysis, such as ﬁnding hypersurfaces with prescribed Hm curvature, see e.g. [12, 21]. In this paper we consider the ﬂow (1.1) with the speed σ given by a power of an mth mean c”
The global curvature estimate for the n-2 Hessian equation (2020, background): “Alexandrov also posed the problem of prescribing general Weingarten curvature on the outer normal [2, 21].”
The Existence of Convex Body with Prescribed Curvature Measures (2009, result): “1) is similar to the equation in [8], where a problem of prescribing Weingarten curvature was considered.”
A class of curvature type equations. (2019, methodology): “orem. 3.1.2. C0 estimate. In viewing of the estimate obtained in Proposition 3.1, the estimate on kD2uk L∞(Sn) will follow from the C0 estimate of u. In general, for the prescribing curvature problem [21] and the Christoﬀel-Minkowski problem [22], the C0 estimates are obtained by using Cheng-Yau’s a”
On the curvature estimates for Hessian equations (2016, background): “ to Caﬀarelli-Nirenberg-Spruck [8]. When f is independent of normal vector ν, C2 estimate has been proved by Caﬀralli-Nirenberg-Spruck [10]. If f in (1.1) depends only on ν, C2 estimate was proved in [18]. Ivochkina [22, 23] considered the Dirichlet problem of equation (1.1) on domains in Rn, C2 est”
Hessian equations with Infinite Dirichlet Boundary Value (2006, background): “See, for example, [2, 5, 6, 16, 18] for Hessian in Ω and [1, 8, 9, 10] for Hessian on the sphere.”
Local estimates for some fully nonlinear elliptic equations (2005, background): “= κ 1 n (x)( v2 +|∇v|2 v2 )n+2 2n . Maximal principles for the ﬁrst equation is studied in Caﬀarelli, Nirenberg, and Spruck [1], and for the latter by B.Guan and J. Spruck [5]. See also [6], [16] and [4] for related works. In this paper, we consider more general operators, which in particular includ”
The Neumann Problem for Hessian Equations (2015, methodology): “n operator. For the curvature equations in classical geometry, the existence of hypersurfaces with prescribedWeingartencurvaturewasstudiedby Pogorelov[34], Caﬀarelli-NirenbergSpruck [3, 4], Guan-Guan [14], Guan-Ma [15] and the later work by Sheng-TrudingerWang [35]. The Hessian equation on Riemannia”
The Christoffel-Minkowski Problem II: Weingarten Curvature Equations* (2006, background): “The prescribing Weingarten curvature problem and similar problems have been studied by various authors, we refer to [3, 17, 16, 14, 5, 6, 18, 8, 9, 10] and references there for related works.”
Liouville property and regularity of a Hessian quotient equation (2003, background): “In [15], the Hessian quotient equation on Sn was studied in connection with a geometric problem of prescribing curvatures.”