Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds
-
Paul Bryan
,
Mohammad N. Ivaki
and
Julian Scheuer
Abstract
We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant nonnegative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a bonus term for mean curvature flow in locally symmetric Riemannian Einstein manifolds of nonnegative sectional curvature. Using a concept of “duality” for strictly convex hypersurfaces, we also obtain a new type of inequality, so-called “pseudo”-Harnack inequality, for expanding flows in the sphere and in the hyperbolic space.
Dedicated to the memory of Peter M. Gruber
Funding statement: The work of the first author was supported in part by the EPSRC on a Programme Grant entitled “Singularities of Geometric Partial Differential Equations” reference number EP/K00865X/1. The work of the second author was supported by Austrian Science Fund (FWF) Project M1716-N25 and the European Research Council (ERC) Project 306445. The work of the third author has been supported by the “Deutsche Forschungsgemeinschaft” (DFG, German research foundation) within the research grant “Harnack inequalities for curvature flows and applications”, grant number SCHE 1879/1-1.
References
[1] B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 (1994), no. 2, 151–171. 10.1007/BF01191340Search in Google Scholar
[2] B. Andrews, Harnack inequalities for evolving hypersurfaces, Math. Z. 217 (1994), no. 2, 179–197. 10.1007/BF02571941Search in Google Scholar
[3] B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. reine angew. Math. 608 (2007), 17–33. 10.1515/CRELLE.2007.051Search in Google Scholar
[4] B. Andrews, X. Chen, H. Fang and J. McCoy, Expansion of co-compact convex spacelike hypersurfaces in Minkowski space by their curvature, Indiana Univ. Math. J. 64 (2015), no. 2, 635–662. 10.1512/iumj.2015.64.5485Search in Google Scholar
[5] A. L. Besse, Einstein manifolds, Ergeb. Math. Grenzgeb. (3) 10, Springer, Berlin 1987. 10.1007/978-3-540-74311-8Search in Google Scholar
[6] P. Bryan and M. N. Ivaki, Harnack estimate for mean curvature flow on the sphere, preprint (2015), https://arxiv.org/abs/1508.02821. 10.4310/AJM.2020.v24.n1.a7Search in Google Scholar
[7] P. Bryan, M. N. Ivaki and J. Scheuer, Harnack inequalities for evolving hypersurfaces on the sphere, Comm. Anal. Geom. 26 (2018), no. 5, 1047–1077. 10.4310/CAG.2018.v26.n5.a2Search in Google Scholar
[8] J. A. Buckland, Short-time existence of solutions to the cross curvature flow on 3-manifolds, Proc. Amer. Math. Soc. 134 (2006), no. 6, 1803–1807. 10.1090/S0002-9939-05-08204-3Search in Google Scholar
[9] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), no. 3–4, 261–301. 10.1007/BF02392544Search in Google Scholar
[10] B. Chow, On Harnack’s inequality and entropy for the Gaussian curvature flow, Comm. Pure Appl. Math. 44 (1991), no. 4, 469–483. 10.1002/cpa.3160440405Search in Google Scholar
[11] B. Chow and R. S. Hamilton, The cross curvature flow of 3-manifolds with negative sectional curvature, Turkish J. Math. 28 (2004), no. 1, 1–10. Search in Google Scholar
[12] F. Fillastre and G. Veronelli, Lorentzian area measures and the Christoffel problem, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 2, 383–467. 10.2422/2036-2145.201405_010Search in Google Scholar
[13] C. Gerhardt, Curvature problems, Ser. Geom. Topol. 39, International Press, Somerville 2006. Search in Google Scholar
[14] C. Gerhardt, Curvature flows in the sphere, J. Differential Geom. 100 (2015), no. 2, 301–347. 10.4310/jdg/1430744123Search in Google Scholar
[15] G. Glaeser, Fonctions composées différentiables, Ann. of Math. (2) 77 (1963), 193–209. 10.2307/1970204Search in Google Scholar
[16] R. S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2, 153–179. 10.4310/jdg/1214440433Search in Google Scholar
[17] R. S. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom. 37 (1993), no. 1, 225–243. 10.4310/jdg/1214453430Search in Google Scholar
[18] R. S. Hamilton, Harnack estimate for the mean curvature flow, J. Differential Geom. 41 (1995), no. 1, 215–226. 10.4310/jdg/1214456010Search in Google Scholar
[19] G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math. 183 (1999), no. 1, 45–70. 10.1007/BF02392946Search in Google Scholar
[20] M. N. Ivaki, Centro-affine normal flows on curves: Harnack estimates and ancient solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 6, 1189–1197. 10.1016/j.anihpc.2014.07.001Search in Google Scholar
[21] M. N. Ivaki, Convex bodies with pinched Mahler volume under the centro-affine normal flows, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 831–846. 10.1007/s00526-014-0807-9Search in Google Scholar
[22] H. Kang and K.-A. Lee, Harnack inequality and pinching estimates for anisotropic curvature flow of hypersurfaces, J. Math. Anal. Appl. 464 (2018), no. 1, 32–57. 10.1016/j.jmaa.2018.03.062Search in Google Scholar
[23] P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3–4, 153–201. 10.1007/BF02399203Search in Google Scholar
[24] Y. Li, Harnack inequality for the negative power Gaussian curvature flow, Proc. Amer. Math. Soc. 139 (2011), no. 10, 3707–3717. 10.1090/S0002-9939-2011-11039-6Search in Google Scholar
[25] D. G. Mead, Newton’s identities, Amer. Math. Monthly 99 (1992), no. 8, 749–751. 10.1080/00029890.1992.11995923Search in Google Scholar
[26] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. 10.1002/cpa.3160170106Search in Google Scholar
[27] K. Nomizu and T. Sasaki, Affine differential geometry. Geometry of affine immersions, Cambridge Tracts in Math. 111, Cambridge University Press, Cambridge 1994. Search in Google Scholar
[28]
G. Pipoli and C. Sinestrari,
Mean curvature flow of pinched submanifolds of
[29] J. Scheuer, Isotropic functions revisited, Arch. Math. (Basel) 110 (2018), no. 6, 591–604. 10.1007/s00013-018-1162-4Search in Google Scholar
[30] K. Smoczyk, Harnack inequalities for curvature flows depending on mean curvature, New York J. Math. 3 (1997), 103–118. Search in Google Scholar
[31]
J. Wang,
Harnack estimate for the
[32] H. Yu, Dual flows in hyperbolic space and de Sitter space, PhD thesis, University of Heidelberg, 2017. Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Linear extension operators between spaces of Lipschitz maps and optimal transport
- Cuspidal ℓ-modular representations of p-adic classical groups
- Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds
- A strong stability condition on minimal submanifolds and its implications
- Stable s-minimal cones in ℝ3 are flat for s ~ 1
- Representation theoretic realization of non-symmetric Macdonald polynomials at infinity
- Brownian motion on Perelman’s almost Ricci-flat manifold
- Hessenberg varieties and hyperplane arrangements
- Limits of canonical forms on towers of Riemann surfaces
Articles in the same Issue
- Frontmatter
- Linear extension operators between spaces of Lipschitz maps and optimal transport
- Cuspidal ℓ-modular representations of p-adic classical groups
- Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds
- A strong stability condition on minimal submanifolds and its implications
- Stable s-minimal cones in ℝ3 are flat for s ~ 1
- Representation theoretic realization of non-symmetric Macdonald polynomials at infinity
- Brownian motion on Perelman’s almost Ricci-flat manifold
- Hessenberg varieties and hyperplane arrangements
- Limits of canonical forms on towers of Riemann surfaces
