Time: Tuesdays, Thursdays at 9:30am.
Rubinstein. Office hours: By appointment.
Room: Mathematics Building 0104.
The goal will be to give an introduction to Geometric Analysis
that is accessible to beginning students interested in
PDE/Analysis or Geometry but not necessarily in both nor
necessarily with background in both. Topics will range, e.g.,
from Calculus of Variations, Bochner technique, Morse theory, weak
solutions and elliptic regularity, maximum principle for elliptic and
parabolic equations, Green's function of the Laplacian, isoperimetric and
Sobolev inequalities, continuity method, curvature and comparison results,
harmonic maps, curvature prescription problems.
Requirement: each student taking the course for a grade will be asked
to prepare and typeset notes for a block of lectures as well as the
solutions of the homework
exercises assigned during those lectures.
Some nonlinear problems in Riemannian geometry, Springer, 1998.
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of
Second Order, Springer, 2001.
P. Li, Geometric Analysis, Cambridge University Press, 2012.
P. Petersen, Riemannian Geometry, Springer, 2006.
M.M. Postnikov, Geometry VI: Riemannian Geometry, Springer, 2001.
R. Schoen, S.-T. Yau, Lectures on Differential Geometry, Int. Press, 1994.
L. Simon, Lectures on PDEs, 2013.
M. Struwe, Variational Methods, 4th Ed., Springer, 2008.
Ambrosio, Gigli, A user's guide to optimal transportation (available
Notes on the construction of Green's function on a Riemannian
Lecture notes from 2013:
Lectures 1-4 (Ryan Hunter)
Lectures 5-6 (Jacky Chong)
Lectures 9-10 (Jason Suagee)
Lectures 11-12 (Siming He)
Lectures 13-15 (Zhenfu Wang)
Lectures 16 & 19 (Siming He)
Lectures 17-18 (Bo Tian)
Lectures 20-21 (Bo Tian)
Overview. Basic definitions of Riemannian geometry. Langrangians
and Euler-Lagrange equations. The length Lagrangian and its
More basic definitions of Riemannian geometry.
Parallel translation and the geodesic equation.
Comparison with the E-L equation from last time.
Jacobi theory I.
Jacobi theory II.
Jacobi theory III.
The direct method in the calculus of variations.
Compactness of sublevel sets as motivation for requiring weak sequential
lower semicontinuity and coercivity.
Situations where the direct method can be applied: p-Laplacian, harmonic
maps into Euclidean space (generalizing geodesics).
Bochner technique, I.
Bochner technique, II, Killing fields.
Bochner technique, III, Stokes' theorem and integration on manifolds.
Bochner technique, IV, application to 1-forms.
The direct method and
Constrained minimization and the direct method.
Cyclical monotonicity. The fundamental theorem of optimal transporatation.
Cyclical monotonicity and Rockafellar's theorem. The fundamental theorem
of optimal transporatation, continued.
Brenier's theorem, I.
Brenier's theorem, II.
Regularity weak solutions of semi-linear equations (L. Simon's lecture 6).
Existence of weak solutions of semi-linear equations, Lax-Milgram Lemma,
and establishing coercivity under ellipticity assumption. Interpolation
(L. Simon's lecture 7).
Spectrum of self-adjoint operators
(L. Simon's lecture 10).
Spectrum of self-adjoint operators - continued.
Parabolic equations, heat kernel
(L. Simon's lecture 11).
Weyl's asymptotic formula (L. Simon's lecture 11).
Main lemma of interior Schauder theory (Schauder estimates via scaling)
(L. Simon's lecture 12).
Schauder estimates: the method of freezing coefficients (L. Simon's
The (weak and strong) Maximum Principle for second order elliptic
The Hopf boundary point lemma (L. Simon's
Outlook and further reading.