MATH 668-TOPICS  IN DIFFERENTIAL GEOMETRY-SPRING 2014-A. FREIRE

W 1/8  Overview/ AAG Sect 3 (start)
[AAG1] [AAG2] [AAG3]

M 1/13  AAG section 3: inner/outer weak solutions (To think about: proof of thm 3.1 (ii) (detail). What is the weak evolution of a disk with two radial
"whiskers" in the plane? Does Gamma depend only on Gamma_0, or also on D_0?)

W 1/15   Derivation of equations for axially symmetric MCF/ Angenent's Sturmian intersection theorem/ stationary solutions
Issues to think about (i) Reduce the proof of thm 4.1 to the Sturmian intersection result (the issue are the boundary conditions)
(ii) Find profiles with H>0 in [0,L] (monotone, with a neck at 0 and a local max at L, with given radii A and B at 0 and L (resp.))

References: S. Angenent: 1. J. Reine angew. Math. 399 (1988) 79-96; 2. Annals 133 (1991) 171-215 (Section 1); JDG 33 (1991) 601-633 (Section 2).

M1/20 MLK Holiday (no lecture)

W 1/22 [AAG] Section 4: Theorems 4.1, 4.3(b); Lemmas 4.4, 4.7

Topics for presentation (i) Section 8 of [AAG] (Caleb); (ii) Section 9 of T. Ilmanen's 1992 Indiana paper.

M 1/27, section 5 of [AAG]

W 1/29, sections 5, 6 of [AAG]
References for asymptotics at a neckpinch: Z. Gang, I.M Sigal, Neck Pinching Dynamics under MCF, J.Geom.Anal. 2009/ Z.Gang, D.Knopf,I.M.Sigal, Neckpinch Dynamics for Asymmetric Surfaces under MCF (Arxiv 1109, see D.Knopf's list)/ Gang &Knopf, Universality in MCF neckpinches, ArXiv 2013 (see D. Knopf's ArXiv list).

M 2/3 Huisken-Sinestrari I (and Huisken 85: decay of Lp norms)/ Discussion of asymptotics (Gang-Sigal)

W 2/5 Hamilton's shrinking dumbbell (Caleb, based on [AAG, sect 8]

M 2/10 Huisken's Sobolev iteration argument for convex MCF (pinching of 2nd ff)

W 2/12 Huisken-Sinestrari I: main theorem (almost-positivity of scalar curvature)

M 2/17 Huisken-Sinestrari II: the induction argument

W 2/19 Huisken-Sinestrari II: type II blowups (Hamilton), weak convexity of limits (see also Mantegazza Ch.4)

M 2/24 Harnack inequality for MCF (Hamilton)

W 2/26 Eternal solutions attaining mean curvature max are translation solitons (Hamilton)
Technical lemmas

M 3/3 Hamilton's theorem on convex hypersurfaces with pinched 2nd ff
Note on a theorem of R. Hamilton

W 3/5 Inverse Mean Curvature Flow (start)[Huisken-Ilmanen]

M 3/10 Weak solutions of IMCF/ BV functions[Huisken-Ilmanen]

W 3/12 Sets of finite perimeter/ Perimeter measure and reduced boundary
(see sections 3 and 4 of the note "Technical lemmas" (posted above)

M 3/17 to F 3/21: SPRING BREAK

M 3/24 Weak solutions of IMCF: properties[Huisken-Ilmanen]

W 3/26 Weak solutions of IMCF: properties/ starshaped hypersurfaces under 1/F flows [Gerhardt]

M  3/31 Evolution of starshaped hypersurfaces under 1/F flows [Gerhardt, Urbas]

W 4/2 Starshaped hypersurfaces under IMCF [Gerhardt, Huisken-Ilmanen]

M 4/7 The Simons cone and smooth minimal surfaces close to it [Velazquez]

W 4/9 Type-two blowup near the Simons cone [Velazquez] (Kevin)

M 4/14 [Andrews]-noncollapsing in mean-convex MCF (Brian)

W 4/16 [JunfangLi] Geometric inequalities for starshaped domains (Josh L.)

M  4/21 Aleksandrov reflection and geometric evolution [Chow-Gulliver]-(Josh M.)

W 4/23 (last day)