An official website of the United States government
Abstract Given a hermitian line bundle on a closed Riemannian manifold , the self‐dual Yang–Mills–Higgs energies are a natural family of functionalsdefined for couples consisting of a section and a hermitian connection ∇ with curvature . While the critical points of these functionals have been well‐studied in dimension two by the gauge theory community, it was shown in [52] that critical points in higher dimension converge as (in an appropriate sense) to minimal submanifolds of codimension two, with strong parallels to the correspondence between the Allen–Cahn equations and minimal hypersurfaces. In this paper, we complement this idea by showing the Γ‐convergence of to (2π times) the codimension two area: more precisely, given a family of couples with , we prove that a suitable gauge invariant Jacobian converges to an integral ‐cycle Γ, in the homology class dual to the Euler class , with mass . We also obtain a recovery sequence, for any integral cycle in this homology class. Finally, we apply these techniques to compare min‐max values for the ‐area from the Almgren–Pitts theory with those obtained from the Yang–Mills–Higgs framework, showing that the former values always provide a lower bound for the latter. As an ingredient, we also establish a Huisken‐type monotonicity result along the gradient flow of .
more »
« less
This content will become publicly available on May 3, 2026
Colding–Minicozzi entropies in Cartan–Hadamard manifolds
Abstract We introduce a family of functionals defined on the set of submanifolds of Cartan–Hadamard manifolds which generalize the Colding–Minicozzi entropy of submanifolds of Euclidean space.We show these functionals are monotone under mean curvature flow under natural conditions.As a consequence, we obtain sharp lower bounds on these entropies for certain closed hypersurfaces and observe a novel rigidity phenomenon.
more »
« less
- PAR ID:
- 10600438
- Publisher / Repository:
- Walter de Gruyter GmbH
- Date Published:
- Journal Name:
- Journal für die reine und angewandte Mathematik (Crelles Journal)
- ISSN:
- 0075-4102
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We study an analog in CR-geometry of the conformal volume of Li–Yau. In particular, to submanifolds of odd-dimensional spheres that are Legendrian or, more generally, horizontal with respect to the sphere’s standard CR-structure, we associate a quantity that is invariant under the CR-automorphisms of the sphere. We apply this concept to a corresponding notion of Willmore energy.more » « less
-
Summary This paper studies nonparametric identification in market-level demand models for differentiated products with heterogeneous consumers. We consider a general class of models that allows for the individual-specific coefficients to vary continuously across the population and give conditions under which the density of these coefficients, and hence also functionals such as the fractions of individuals who benefit from a counterfactual intervention, is identified.more » « less
-
null (Ed.)A bstract While the study of bordered (pseudo-)holomorphic curves with boundary on Lagrangian submanifolds has a long history, a similar problem that involves (special) Lagrangian submanifolds with boundary on complex surfaces appears to be largely overlooked in both physics and math literature. We relate this problem to geometry of coassociative submanifolds in G 2 holonomy spaces and to Spin(7) metrics on 8-manifolds with T 2 fibrations. As an application to physics, we propose a large class of brane models in type IIA string theory that generalize brane brick models on the one hand and 2d theories T [ M 4 ] on the other.more » « less
