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1. Min–max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications

   https://doi.org/10.1007/s00208-020-02096-0  

   Guang, Qiang; Li, Martin Man-chun; Wang, Zhichao; Zhou, Xin (April 2021, Mathematische Annalen)

   null (Ed.)

   Abstract For any smooth Riemannian metric on an $$(n+1)$$ ( n + 1 ) -dimensional compact manifold with boundary $$(M,\partial M)$$ ( M , ∂ M ) where $$3\le (n+1)\le 7$$ 3 ≤ ( n + 1 ) ≤ 7 , we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min–max theory in the Almgren–Pitts setting. We apply our Morse index estimates to prove that for almost every (in the $$C^\infty $$ C ∞ Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M . If $$\partial M$$ ∂ M is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting. 

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2. Existence of hypersurfaces with prescribed mean curvature I – generic min-max

   Zhou, Xin; Zhu, Jonathan J. (April 2020, Cambridge journal of mathematics)

   We prove that, for a generic set of smooth prescription functions h on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature h. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersurface of multiplicity one. More precisely, we show that our previous min-max theory, developed for constant mean curvature hypersurfaces, can be extended to construct min-max prescribed mean curvature hypersurfaces for certain classes of prescription function, including a generic set of smooth functions, and all nonzero analytic functions. In particular we do not need to assume that h has a sign. 

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3. Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation

   https://doi.org/10.1002/num.23017  

   Doan, Cao‐Kha; Hoang, Thi‐Thao‐Phuong; Ju, Lili (September 2023, Numerical Methods for Partial Differential Equations)

   Abstract The Allen‐Cahn equation satisfies the maximum bound principle, that is, its solution is uniformly bounded for all time by a positive constant under appropriate initial and/or boundary conditions. It has been shown recently that the time‐discrete solutions produced by low regularity integrators (LRIs) are likewise bounded in the infinity norm; however, the corresponding fully discrete error analysis is still lacking. This work is concerned with convergence analysis of the fully discrete numerical solutions to the Allen‐Cahn equation obtained based on two first‐order LRIs in time and the central finite difference method in space. By utilizing some fundamental properties of the fully discrete system and the Duhamel's principle, we prove optimal error estimates of the numerical solutions in time and space while the exact solution is only assumed to be continuous in time. Numerical results are presented to confirm such error estimates and show that the solution obtained by the proposed LRI schemes is more accurate than the classical exponential time differencing (ETD) scheme of the same order. 

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4. Convergence of the self‐dual U (1)‐Yang–Mills–Higgs energies to the (n−2)$(n-2)$‐area functional

   https://doi.org/10.1002/cpa.22150  

   Parise, Davide; Pigati, Alessandro; Stern, Daniel (January 2024, Communications on Pure and Applied Mathematics)

   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 . 

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5. The p-widths of a surface

   https://doi.org/10.1007/s10240-023-00141-7  

   Chodosh, Otis; Mantoulidis, Christos (June 2023, Publications mathématiques de l'IHÉS)

   Abstract The $$p$$ p -widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the $$p$$ p -widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets. We then prove optimality of the sweepouts of the round two-sphere constructed from the zero set of homogeneous polynomials, showing that the $$p$$ p -widths of the round sphere are attained by $$\lfloor \sqrt{p}\rfloor $$ ⌊ p ⌋ great circles. As a result, we find the universal constant in the Liokumovich–Marques–Neves–Weyl law for surfaces to be $$\sqrt{\pi }$$ π . En route to calculating the $$p$$ p -widths of the round two-sphere, we prove two additional new results: a bumpy metrics theorem for stationary geodesic nets with fixed edge lengths, and that, generically, stationary geodesic nets with bounded mass and bounded singular set have Lusternik–Schnirelmann category zero. 

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