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1. Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates

   https://doi.org/10.4007/annals.2020.191.1.4  

   Chodosh, Otis; Mantoulidis, Christos (January 2020, Annals of mathematics)

   The Allen–Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang–Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show that for generic metrics on a 3-manifold, minimal surfaces arising from Allen–Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen–Cahn setting, a strong form of the multiplicity one-conjecture and the index lower bound conjecture of Marques–Neves in 3-dimensions regarding min-max constructions of minimal surfaces. Allen–Cahn min-max constructions were recently carried out by Guaraco and Gaspar–Guaraco. Our resolution of the multiplicity-one and the index lower bound conjectures shows that these constructions can be applied to give a new proof of Yau's conjecture on infinitely many minimal surfaces in a 3-manifold with a generic metric (recently proven by Irie–Marques–Neves) with new geometric conclusions. Namely, we prove that a 3-manifold with a generic metric contains, for every p = 1, 2, 3,…, a two-sided embedded minimal surface with Morse index p and area ~ p13, as conjectured by Marques-Neves. 

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2. Half-Space Intersection Properties for Minimal Hypersurfaces

   https://doi.org/10.1007/s12220-025-02006-3  

   Naff, Keaton; Zhu, Jonathan J (June 2025, The Journal of Geometric Analysis)

   We prove ``half-space" intersection properties in three settings: the hemisphere, half-geodesic balls in space forms, and certain subsets of Gaussian space. For instance, any two embedded minimal hypersurfaces in the sphere must intersect in every closed hemisphere. Two approaches are developed: one using classifications of stable minimal hypersurfaces, and the second using conformal change and comparison geometry for $$\alpha$$-Bakry-\'{E}mery-Ricci curvature. Our methods yield the analogous intersection properties for free boundary minimal hypersurfaces in space form balls, even when the interior or boundary curvature may be negative. Finally, Colding and Minicozzi recently showed that any two embedded shrinkers of dimension $$n$$ must intersect in a large enough Euclidean ball of radius $R(n)$. We show that $$R(n) \leq 2 \sqrt{n}$$. 

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3. The half-volume spectrum of a manifold

   https://doi.org/10.1007/s00526-025-02949-z  

   Mazurowski, Liam; Zhou, Xin (May 2025, Calculus of Variations and Partial Differential Equations)

   Abstract We define the half-volume spectrum$$\{{\tilde{\omega }_p\}_{p\in \mathbb {N}}}$$ $\{{\tilde{\omega }}_p{\}}_{p\in N}$of a closed manifold$$(M^{n+1},g)$$ $(M^{n+1},g)$. This is analogous to the usual volume spectrum ofM, except that we restrict top-sweepouts whose slices each enclose half the volume ofM. We prove that the Weyl law continues to hold for the half-volume spectrum. We define an analogous half-volume spectrum$$\tilde{c}(p)$$ $\tilde{c}\left(p\right)$in the phase transition setting. Moreover, for$$3 \le n+1 \le 7$$ $3\le n+1\le 7$, we use the Allen–Cahn min-max theory to show that each$$\tilde{c}(p)$$ $\tilde{c}\left(p\right)$is achieved by a constant mean curvature surface enclosing half the volume ofMplus a (possibly empty) collection of minimal surfaces with even multiplicities. 

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4. Quantitative Rigidity of Differential Inclusions in Two Dimensions

   https://doi.org/10.1093/imrn/rnad108  

   Lamy, Xavier; Lorent, Andrew; Peng, Guanying (May 2023, International Mathematics Research Notices)

   Abstract For any compact connected one-dimensional submanifold $$K\subset \mathbb R^{2\times 2}$$ without boundary that has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate $$\begin{align*} \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\, \textrm{d}x \leq C \int_{B_1} \operatorname{dist}^2(Du, K)\, \textrm{d}x, \qquad\forall u\in H^1(B_1;\mathbb R^2). \end{align*}$$This is an optimal generalization, for compact connected submanifolds of $$\mathbb R^{2\times 2}$$ without boundary, of the celebrated quantitative rigidity estimate of Friesecke, James, and Müller for the approximate differential inclusion into $SO(n)$. The proof relies on the special properties of elliptic subsets $$K\subset{{\mathbb{R}}}^{2\times 2}$$ with respect to conformal–anticonformal decomposition, which provide a quasilinear elliptic partial differential equation satisfied by solutions of the exact differential inclusion $$Du\in K$$. We also give an example showing that no analogous result can hold true in $$\mathbb R^{n\times n}$$ for $$n\geq 3$$. 

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5. Local version of Courant’s nodal domain theorem

   https://doi.org/10.4310/jdg/1707767334  

   Chanillo, Sagun; Logunov, Alexander; Malinnikova, Eugenia; Mangoubi, Dan (January 2024, Journal of Differential Geometry)

   Let (M,g) be a compact n-dimensional Riemannian manifold without boundary, where the metric g is C^1-smooth. Consider the sequence of eigenfunctions u_k of the Laplace operator on M. Let B be a ball on M. We prove that the number of nodal domains of u_k that intersect B is not greater than C_1Volume(B)k+C_2k^{(n-1)/n}, where C_1 and C_2 depend on (M,g) only. The problem of local bounds for the volume and for the number of nodal domains was raised by Donnelly and Fefferman, who also proposed an idea how one can prove such bounds. We combine their idea with two ingredients: the recent sharp Remez type inequality for eigenfunctions and the Landis type growth lemma in narrow domains. 

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