Afuni, Ahmad Local energy inequalities for mean curvature flow into evolving ambient spaces. (English) Zbl 1435.53065 Manuscr. Math. 158, No. 3-4, 317-343 (2019). Summary: We establish a local monotonicity formula for mean curvature flow into a curved space whose metric is also permitted to evolve simultaneously with the flow, extending the work of K. Ecker [Ann. Math. (2) 154, No. 2, 503–525 (2001; Zbl 1007.53050)], G. Huisken [J. Differ. Geom. 31, No. 1, 285–299 (1990; Zbl 0694.53005)], J. Lott [Commun. Math. Phys. 313, No. 2, 517–533 (2012; Zbl 1246.53090)], A. Magni et al. [J. Evol. Equ. 13, No. 3, 561–576 (2013; Zbl 1275.35118)] and K. Ecker et al. [J. Reine Angew. Math. 616, 89–130 (2008; Zbl 1170.35050)]. This formula gives rise to a monotonicity inequality in the case where the target manifold’s geometry is suitably controlled, as well as in the case of a gradient shrinking Ricci soliton. Along the way, we establish suitable local energy inequalities to deduce the finiteness of the local monotone quantity. MSC: 53E10 Flows related to mean curvature 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35K55 Nonlinear parabolic equations 58J35 Heat and other parabolic equation methods for PDEs on manifolds 58C99 Calculus on manifolds; nonlinear operators Citations:Zbl 1007.53050; Zbl 0694.53005; Zbl 1246.53090; Zbl 1275.35118; Zbl 1170.35050 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Afuni, A.: Heat ball formulæ for \[k\] k-forms on evolving manifolds. Advances in Calculus of Variations. https://doi.org/10.1515/acv-2017-0026 · Zbl 1394.35224 [2] Allard, W .K.: On the first variation of a varifold. Ann. Math. (2) 95, 417-491 (1972) · Zbl 0252.49028 · doi:10.2307/1970868 [3] Brakke, K.A.: The Motion of a Surface by Its Mean Curvature. Mathematical Notes, vol. 20. Princeton University Press, Princeton (1978) · Zbl 0386.53047 [4] Ecker, K.: A local monotonicity formula for mean curvature flow. Ann. Math. (2) 154(2), 503-525 (2001) · Zbl 1007.53050 · doi:10.2307/3062105 [5] Ecker, K.: Regularity Theory for Mean Curvature Flow. Progress in Nonlinear Differential Equations and their Applications, 57. Birkhäuser Boston, Inc., Boston (2004) · Zbl 1058.53054 [6] Ecker, K., Knopf, D., Ni, L., Topping, P.: Local monotonicity and mean value formulas for evolving Riemannian manifolds. J. Reine Angew. Math. 616, 89-130 (2008) · Zbl 1170.35050 [7] Hamilton, R.S.: A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1(1), 113-126 (1993) · Zbl 0799.53048 · doi:10.4310/CAG.1993.v1.n1.a6 [8] Hamilton, R .S.: Monotonicity formulas for parabolic flows on manifolds. Commun. Anal. Geom. 1(1), 127-137 (1993) · Zbl 0779.58037 · doi:10.4310/CAG.1993.v1.n1.a7 [9] Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31(1), 285-299 (1990) · Zbl 0694.53005 · doi:10.4310/jdg/1214444099 [10] Lott, J.: Mean curvature flow in a Ricci flow background. Commun. Math. Phys. 313(2), 517-533 (2012) · Zbl 1246.53090 · doi:10.1007/s00220-012-1503-2 [11] Magni, A., Mantegazza, C., Tsatis, E.: Flow by mean curvature inside a moving ambient space. J. Evol. Equ. 13(3), 561-576 (2013) · Zbl 1275.35118 · doi:10.1007/s00028-013-0190-6 [12] Mullins, W .W.: Two-dimensional motion of idealized grain boundaries. J. Appl. Phys. 27(8), 900-904 (1956). https://doi.org/10.1063/1.1722511 · doi:10.1063/1.1722511 [13] Petersen, P.: Riemannian Geometry. Graduate Texts in Mathematics, vol. 171, 2nd edn. Springer, New York (2006) · Zbl 1220.53002 [14] Poor, W.A.: Differential Geometric Structures. McGraw-Hill Book Company Inc., New York (1981) · Zbl 0493.53027 [15] Warner, F. W.: Foundations of Differentiable Manifolds and Lie Groups, 94 of Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin. Corrected reprint of the 1971 edition (1983) [16] Watson, N .A.: A theory of subtemperatures in several variables. Proc. Lond. Math. Soc. (3) 26, 385-417 (1973) · Zbl 0253.35045 · doi:10.1112/plms/s3-26.3.385 [17] White, B.: A local regularity theorem for mean curvature flow. Ann. Math. (2) 161(3), 1487-1519 (2005) · Zbl 1091.53045 · doi:10.4007/annals.2005.161.1487 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.