Motion of hypersurfaces by Gauss curvature

By Ben Andrews. 2000

186 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/e92d5480d0835a7630881c8a5f78b4679f069890

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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, background): “In some case it is known that convex hypersurfaces shrink to a point [7, 32], but some counterexamples show that in general the profile may not become spherical after rescaling.”
Asymptotic behavior of flows by powers of the Gaussian curvature (2016, methodology): “This was proved by Andrews [3] for α ∈ [ 1 n+2 , 1 n ], by Guan and Ni [13] for α = 1, by Kim and 1”
A unified flow approach to smooth, even Lp-Minkowski problems (2016, background): “Also, in higher dimensions, the existence of solutions to the Lp-Minkowski problems follows from [11] when −n− 1 < p ≤ −n+ 1 (a short proof of this is also given in [38]) or when φ is even (e.”
Entropy and a convergence theorem for Gauss curvature flow in high dimension (2013, methodology): “A similar consideration was applied by Andrews in [6] to construct solitons of the flow with the speed being the power of the Gauss curvature.”
Evolution of convex hypersurfaces by powers of the mean curvature (2005, background): “Andrews [2] was able to strengthen this result, showing that for α ∈ (1/(n+2), 1/n) the ratio of the biggest to the smallest principal curvature remains bounded and the surfaces converge after blow-up to a smooth limiting surface.”
Survey on Affine Spheres (2008, background): “Andrews [3] also shows that arbitrary compact convex initial hypersurfaces are instantaneously regularized under the affine normal flow.”
On the uniqueness of Lp-Minkowski problems: The constant p-curvature case in R3 (2015, methodology): “One may consult Andrews [1, 2, 3], Urbas [42, 43], and the references therein for related works in this direction.”
Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems (2017, background): “ y= u(x)x+∇u(x), where ∇ is the covariant derivative with respect to the standard metric eij of the sphere Sn. Hence (2.3) r= |y| = p u2 +|∇u|2. The second fundamental form of M is given by, see e.g. [5, 41], (2.4) hij = uij +ueij, where uij = ∇2 ijudenotes the second order covariant derivative of u”