Pinching estimates and motion of hypersurfaces by curvature functions
By Ben Andrews. 2004
179 citations recorded by the source corpus.
Abstract Second derivative pinching estimates are proved for a class of parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply convergence of convex hypersurfaces to spheres under these flows, improving earlier results of B. Chow and the author. The result is obtained via a detailed analysis of gradient terms in the equations satisfied by second derivatives.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/87db014f7a0120843b09fa9d880a44775b4ac738
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- A microscopic convexity principle for nonlinear partial differential equations (2008, unknown): “ollary 1.3. 4 BAOJUN BIAN AND PENGFEI GUAN We now discuss some geometric equations on general manifolds. Preservation of convexity is an important issue for the geometric flows of hypersurfaces (e.g., [21, 5] and references therein). We have the following general result. Theorem 1.4. Suppose F(A,X,~n”
- A constant rank theorem for solutions of fully nonlinear elliptic equations (2007, methodology): “The same condition, together with some proper convex cone condition on and concavity condition on f , were also used by Andrews in [2] on pinching estimates of evolving closed convex hypersurfaces in Rn+1.”
- Mixed volume preserving curvature flows (2005, methodology): “Since F is inverse concave, Theorem 4.1 from [ A4 ] gives that at the null eigenvector, � ¨”
- Curvature flows in the sphere (2013, unknown): “Andrews therefore proved in [4, Theorem 3.2] that the conclusion is still valid if ˜ N ij satisfies the weaker condition (5.17) ˜ N ij η i η j + sup Moreover, he proved that the weaker condition is satisfied by the present tensor ˜ N ij , cf. [4, Theorem 4.1], provided ˜ F and F are both concave, cf.…”
- Hyperbolic Alexandrov-Fenchel quermassintegral inequalities I (2013, background): “7) was proved in [15] with the help of a maximal principle for tensors of Andrews [1] .”
- Quermassintegral preserving curvature flow in Hyperbolic space (2017, methodology): “ch argument, and inverse-concave, and QUERMASSINTEGRAL PRESERVING CURVATURE FLOW IN HYPERBOLIC SPACE 7 G2 satisfies Assumption 1.1, then F= Gσ 1G 1−σ 2 satisfies Assumption 1.1for any 0 <σ<1 (see [4, 6] for more examples of inverse concave or convex functions). 2.3. Evolution equations. Along th”
- Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space (2006, unknown): “This result was, to our knowledge, first stated in [1]; for proofs one may consult [11, 2 ].”
- On a class of fully nonlinear flows in Kähler geometry (2009, methodology): “We use a nice trick due to Andrews [1].”
- Locally constrained curvature flows and geometric inequalities in hyperbolic space (2020, methodology): “As in [2], the quotient Em (κ)/Em−1(κ) is both concave and inverse concave.”
- Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere (2013, methodology): “The most important examples of curvature functions F being concave and inverse concave are (cid:16) (cid:17) P For a proof of the inverse concavity of these functions see the proofs of [2, Theorem 2.6, Theorem 2.7].”
- Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space (2013, methodology): “This can be proved by using Andrews’ pinching estimates [1].”