Student Analysis Seminar (Spring 2025): Min-max theory for minimal submanifolds
Student Analysis Seminar (Spring 2025): Min-max theory for minimal submanifolds
Organizers: Raphael Tsiamis, Yipeng Wang
Time: Tuesdays 4 - 5:20 pm
Location: Columbia Math Department, Room 507
This seminar will study the construction of minimal submanifolds via min-max procedures in the calculus of variations. We will begin by reviewing the relevant tools from geometric measure theory (stationary varifolds, Allard's regularity theorem) corresponding to critical points of the area functional.
We will first review some of the ideas of Simon-Smith min-max theory. Then, we will turn our attention to new variational methods aimed at building minimal submanifolds of higher codimension in a closed ambient Riemannian manifold by means of min-max procedures. We will also be interested in the minimal surfaces and minimal submanifolds of codimension 2 obtained using the variational theory of a suitable relaxation of the area functional. This will involve studying the properties of the Allen-Cahn functional, abelian Higgs model, Ginzburg-Landau energy, and the Yang-Mills energy, which have recently led to important breakthroughts in differential geometry and the calculus of variations.
Throughout the seminar, our focus will be on obtaining the multiplicity, regularity, and analyticity of the minimizers to such functionals in the calculus of variations. We will also discuss estimates on the dimension of the singular set of the resulting critical object (varifold).
O. Chodosh, C. Mantoulidis. Minimal surfaces and the Allen-Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates. Ann. of Math. (2) 191(1): 213-328, 2020.
M. Guaraco. Min–max for phase transitions and the existence of embedded minimal hypersurfaces. J. Differential Geom., 108(1):91–133, 2018.
P. Gaspar, M. Guaraco. The Allen-Cahn equation on closed manifolds. Calc. Var. Partial Differ. Equ., 57(4):Art. 101, 42, 2018.
F. Pacard. The role of minimal surfaces in the study of the Allen-Cahn equation. In Geometric analysis: partial differential equations and surfaces, Volume 570 of Contemp. Math., pages 137–163. Amer. Math. Soc., Providence, RI, 2012.
A. Pigati, T. Rivière. A proof of the multiplicity one conjecture for min-max minimal surfaces in arbitrary codimension. Duke Math. J. 169(11): 2005-2044, 2020.
A. Pigati, D. Stern. Minimal submanifolds from the abelian Higgs model. Invent. Math. 223: 1027-1095, 2021.
K. Wang, J. Wei, Finite Morse index implies finite ends. Comm. Pure Appl. Math., 72(5):1044– 1119, 2019.
In the first meeting, we will set out the goals of the seminar and discuss various aspects of the references, schedule, and content of the talks.
In the first meeting, we will set out the goals of the seminar and review the main tools of our theory. Specifically, we will review various key notions from geometric measure theory, particularly rectifiable varifolds and their properties. We will then introduce Allard's regularity theorem for almost stationary varifolds and discuss some important aspects and techniques required for its proof. The emphasis will be on techniques related to the excess and its decay that occur throughout the regularity theory of min-max schemes. Finally, we will discuss some of the main applications of this theorem for the regularity of stationary varifolds obtained via min-max procedures.
We continue our discussion of the min-max theory introduced by Simon and Smith. These techniques refined the ideas used by Almgren and Pitts to prove that any closed n-manifold, 3 ≤ n ≤ 7, admits a closed, embedded minimal hypersurface. Their work, applicable to the case n=3, is the basis of many modern constructions in min-max theory. Apart from minimal surfaces, min-max arguments have been used in other contexts, namely the proof of the Willmore conjecture by Marques and Neves and the construction of surfaces with constant mean curvature, prescribed mean curvature, possibly with free or capillary boundary.
We continue our discussion of the min-max theory introduced by Simon and Smith. These techniques refined the ideas used by Almgren and Pitts to prove that any closed n-manifold, 3 ≤ n ≤ 7, admits a closed, embedded minimal hypersurface. Their work, applicable to the case n=3, is the basis of many modern constructions in min-max theory. Apart from minimal surfaces, min-max arguments have been used in other contexts, namely the proof of the Willmore conjecture by Marques and Neves and the construction of surfaces with constant mean curvature, prescribed mean curvature, possibly with free or capillary boundary.
I will discuss the existence and regularity of critical points of the area functional among Lagrangian surfaces in symplectic 4-manifolds, reviewing classical results by R. Schoen and J. Wolfson (existence and partial regularity through the mapping approach) and recent progress by A. Pigati and T. Rivière (within the framework of parametrized varifolds). If time permits, I will also present a recent result, obtained in collaboration with G. Orriols and T. Rivière, concerning the existence of Hamiltonian stationary Lagrangian immersions with multiple singularities.
I will discuss Liouville theorems for minimizers of the Allen-Cahn energy and their connection to global minimal surfaces.
We will introduce the Allen-Cahn equation, satisfied by the minimizers of the Allen-Cahn energy. This energy describes the process of phase separation and transition in various physical systems. We will first obtain some basic properties of this equation and its solutions, including the notion of Γ-convergence, the foundational work of Modica and Mortola, and the Modica bound for solutions to the Allen-Cahn equation. Next, we will introduce the work of Pacard and Ritoré on the connections between constant mean curvature surfaces and the Allen-Cahn equation.
Spring Break recess.
We show that stable cones for the one-phase free boundary problem are hyperplanes in dimension 4. As a corollary, both one and two-phase energy minimizing hypersurfaces are smooth in dimension 4. This is joint work with David Jerison.
We prove the Modica bound, which first appeared in
L. Modica. A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math. 38 (1985) 679–684,
for bounded entire solutions of semilinear equations. We then discuss particular applications of this bound to the Allen-Cahn equation and to the Γ-convergence of sequences of uniformly bounded minimizers of the Allen-Cahn energy.
We discuss the rigidity of stable cones for the one-phase free boundary problem in four dimensions. This is joint work with David Jerison.