A local monotonicity formula for mean curvature flow. (English) Zbl 1007.53050

Ecker, Klaus

Ann. Math. (2) 154, No. 2, 503-525 (2001).

Let \((M_t), t \in (a,b)\) be finite dimensional submanifolds of \(\operatorname{Re}^{n+k}\). A mean curvature flow is established through immersions \(x_t= x(\cdot,t): M \rightarrow \operatorname{Re}^{n+k}\) such that the image \(M_t\) of the manifold satisfies the evolution equation: \[ {\partial x \over \partial t} = \vec{H},\tag{1} \] where \(\vec{H}\) denotes the mean curvature vector of \(M_t\) at \(x(p,t)\) for \((p,t) \in M \times (a,b)\).
In this article a local monotonicity formula is established for the mean curvature flow, motivated by a mean value formula for solutions of the heat equation, due to N. A. Watson [Proc. Lond. Math. Soc., III. Ser. 26, 385-417 (1973; Zbl 0253.35045)]. The author observes that Equation (1) is a weakly parabolic equation drawing ideas from minimal submanifolds; these turn out as stationary solutions of the flow.
The author had previously obtained log-Sobolev inequalities [J. Reine Angew. Math. 522, 105-118 (2000; Zbl 0952.46021)] to derive an integral bound for the heat kernel of a submanifold which is a solution of the mean curvature evolution Equation (1) defined by L. Gross [J. Funct. Anal. 102, No. 2, 268-313 (1991; Zbl 0742.22003)].
This paper is well written, and includes a valuable bibliography. For other mean curvature flow equation studies see U. F. Mayer [Exp. Math. 10, No. 1, 103-107 (2001; Zbl 0982.53061)] and E. De Giorgi [Conjectures on limits of some quasilinear parabolic equations and flow by mean curvature, Pitman Res. Notes Math. Ser. 269, 85-95 (1992; Zbl 0802.35063)].

Reviewer: L.Del Riego (San Luis Potosí)

MSC:

53C44	Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
58J53	Isospectrality
58D25	Equations in function spaces; evolution equations
58C99	Calculus on manifolds; nonlinear operators

Keywords:

heat equation solution estimate; mean curvature flow

Citations:

Zbl 0253.35045; Zbl 0952.46021; Zbl 0742.22003; Zbl 0982.53061; Zbl 0802.35063