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

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)].

### 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
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