Harnack inequality for the negative power Gaussian curvature flow

Li, Yi

doi:10.1090/S0002-9939-2011-11039-6

Harnack inequality for the negative power Gaussian curvature flow
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Proc. Amer. Math. Soc. 139 (2011), 3707-3717 Request permission

Abstract:

In this paper, we study the power of Gaussian curvature flow of a compact convex hypersurface and establish its Harnack inequality when the power is negative. In the Harnack inequality, we require that the absolute value of the power is strictly positive and strictly less than the inverse of the dimension of the hypersurface.

References

Ben Andrews, Harnack inequalities for evolving hypersurfaces, Math. Z. 217 (1994), no. 2, 179–197. MR 1296393, DOI 10.1007/BF02571941
Bennett Chow, Deforming convex hypersurfaces by the $n$th root of the Gaussian curvature, J. Differential Geom. 22 (1985), no. 1, 117–138. MR 826427
Bennett Chow, On Harnack’s inequality and entropy for the Gaussian curvature flow, Comm. Pure Appl. Math. 44 (1991), no. 4, 469–483. MR 1100812, DOI 10.1002/cpa.3160440405
Richard S. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom. 37 (1993), no. 1, 225–243. MR 1198607
Richard S. Hamilton, Harnack estimate for the mean curvature flow, J. Differential Geom. 41 (1995), no. 1, 215–226. MR 1316556
Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI 10.1007/BF02399203
Qi-Rui Li, Surfaces expanding by the power of the Gauss curvature flow, Proc. Amer. Math. Soc. 138 (2010), no. 11, 4089–4102. MR 2679630, DOI 10.1090/S0002-9939-2010-10431-8
Oliver C. Schnürer, Surfaces expanding by the inverse Gaußcurvature flow, J. Reine Angew. Math. 600 (2006), 117–134. MR 2283800, DOI 10.1515/CRELLE.2006.088
Knut Smoczyk, Harnack inequalities for curvature flows depending on mean curvature, New York J. Math. 3 (1997), 103–118. MR 1480081
Jie Wang, Harnack estimate for the $H^k$-flow, Sci. China Ser. A 50 (2007), no. 11, 1642–1650. MR 2390477, DOI 10.1007/s11425-007-0095-3