Publications

Journal Articles

2023

1. Min-max theory for free boundary G-invariant minimal hypersurfaces

   Tongrui Wang

   Advances in Mathematics, 2023

   Abs arXiv HTML

   Given a compact Riemannian manifold M^n+1 with dimension 3≤n+1≤7 and ∂M≠\emptyset, the free boundary min-max theory built by Li-Zhou shows the existence of a smooth almost properly embedded minimal hypersurface with free boundary in M. In this paper, we generalize their constructions into equivariant settings. Specifically, let G be a compact Lie group acting as isometries on M with cohomogeneity at least 3. Then there exists a nontrivial smooth almost properly embedded free boundary G-invariant minimal hypersurface. Moreover, if the Ricci curvature of M is non-negative and ∂M is strictly convex, then there exist infinitely many properly embedded G-invariant minimal hypersurfaces with free boundary.

2. Equivariant Morse index of min-max G -invariant minimal hypersurfaces

   Tongrui Wang

   Mathematische Annalen, 2023

   Abs arXiv HTML

   For a closed Riemannian manifold $M^n+1 with a compact Lie group G acting as isometries, the equivariant min–max theory gives the existence and the potential abundance of minimal G-invariant hypersurfaces provided 3\le \textrmcodim(G⋅p) \le 7 for all p∈M. In this paper, we show a compactness theorem for these min–max minimal G-hypersurfaces and construct a G-invariant Jacobi field on the limit. Combining with an equivariant bumpy metrics theorem, we obtain a C^∞_G-generic finiteness result for min–max G-hypersurfaces with area uniformly bounded. As a main application, we further generalize the Morse index estimates for min–max minimal hypersurfaces to the equivariant setting. Namely, the closed G-invariant minimal hypersurface Σ⊂M constructed by the equivariant min–max on a k-dimensional homotopy class can be chosen to satisfy \textrmIndex_G(Σ)\le k$.

2022

1. Min–Max Theory for G-Invariant Minimal Hypersurfaces

   Tongrui Wang

   The Journal of Geometric Analysis, 2022

   Abs arXiv HTML

   In this paper, we consider a closed Riemannian manifold M^n+1 with dimension 3≤n+1≤7, and a compact Lie group G acting as isometries on M with cohomogeneity at least 3. After adapting the Almgren–Pitts min–max theory to a G-equivariant version, we show the existence of a non-trivial closed smooth embedded G-invariant minimal hypersurface Σ⊂M provided that the union of non-principal orbits forms a smooth embedded submanifold of M with dimension at most n-2. Moreover, we also build upper bounds as well as lower bounds of (G, p)-widths, which are analogs of the classical conclusions derived by Gromov and Guth. An application of our results combined with the work of Marques–Neves shows the existence of infinitely many G-invariant minimal hypersurfaces when Ric_M>0 and orbits satisfy the same assumption above.

2. The existence of G-invariant constant mean curvature hypersurfaces

   Tongrui Wang, and Zhiang Wu

   Calculus of Variations and Partial Differential Equations, 2022

   Abs arXiv HTML

   In this paper, we consider a closed Riemannian manifold M^n+1 with dimension 3≤n+1≤7, and a compact Lie group G acting as isometries on M with cohomogeneity at least 3. Suppose the union of non-principal orbits M∖M^reg is a smooth embedded submanifold of M with dimension at most n-2. Then for any c∈\mathbbR, we show the existence of a nontrivial, smooth, closed, almost embedded, G-invariant hypersurface Σ^n of constant mean curvature c.

Preprints

2024

1. Minimal surfaces with low genus in lens spaces

   Tongrui Wang Xingzhe Li, and Xuan Yao

   arXiv preprint arXiv:2406.12584, 2024

   Abs arXiv

   Given a Riemannian ℝℙ3 with a bumpy metric or a metric of positive Ricci curvature, we show that there either exist four distinct minimal real projective planes, or exist one minimal real projective plane together with two distinct minimal 2-spheres. Our proof is based on a variant multiplicity one theorem for the Simon-Smith min-max theory under certain equivariant settings. In particular, we show under the positive Ricci assumption that ℝℙ3 contains at least four distinct minimal real projective planes and four distinct minimal tori. Additionally, the number of minimal tori can be improved to five for a generic positive Ricci metric on ℝℙ3 by the degree method. Moreover, using the same strategy, we show that in the lens space L(4m,2m±1), m≥1, with a bumpy metric or a metric of positive Ricci curvature, there either exist N(m) numbers of distinct minimal Klein bottles, or exist one minimal Klein bottle and three distinct minimal 2-spheres, where N(1)=4, N(m)=2 for m≥2, and the first case happens under the positive Ricci assumption.

2023

1. Equivariant min-max hypersurface in G -manifolds with positive Ricci curvature

   Tongrui Wang

   arXiv preprint arXiv:2304.03656, 2023

   Abs arXiv

   In this paper, we consider a connected orientable closed Riemannian manifold M^n+1 with positive Ricci curvature. Suppose $G is a compact Lie group acting by isometries on M with 3≤codim(G⋅p)≤7 for all p∈M. Then we show the equivariant min-max G-hypersurface Σcorresponding to the fundamental class [M] is a multiplicity one minimal G-hypersurface with a G-invariant unit normal and G-equivariant index one. As an application, we are able to establish a genus bound for Σ, a control on the singular points of Σ/G, and an upper bound for the (first) G-width of M provided n+1=3 and the actions of G$ are orientation preserving.

2. Curvature estimates for stable free boundary minimal hypersurfaces in locally wedge-shaped manifolds

   Liam Mazurowski, and Tongrui Wang

   arXiv preprint arXiv:2307.12948, 2023

   Abs arXiv

   In this paper, we consider locally wedge-shaped manifolds, which are Riemannian manifolds that are allowed to have both boundary and certain types of edges. We define and study the properties of free boundary minimal hypersurfaces inside locally wedge-shaped manifolds. In particular, we show a compactness theorem for free boundary minimal hypersurfaces with curvature and area bounds in a locally wedge-shaped manifold. Additionally, using Schoen-Simon-Yau’s estimates, we also prove a Bernstein-type theorem indicating that, under certain conditions, a stable free boundary minimal hypersurface inside a Euclidean wedge must be a portion of a hyperplane. As our main application, we establish a curvature estimate for sufficiently regular free boundary minimal hypersurfaces in a locally wedge-shaped manifold. We expect this curvature estimate will be useful for establishing a min-max theory for the area functional in wedge-shaped spaces.

3. Min-max theory for free boundary minimal hypersurfaces in locally wedge-shaped manifolds

   Liam Mazurowski, and Tongrui Wang

   arXiv preprint arXiv:2307.12953, 2023

   Abs arXiv

   We develop a min-max theory for the area functional in the class of locally wedge-shaped manifolds. Roughly speaking, a locally wedge-shaped manifold is a Riemannian manifold that is allowed to have both boundary and certain types of edges. Fix a dimension 3 \le n+1 \le 6. As our main theorem, we prove that every compact locally wedge-shaped manifold M^n+1 with acute wedge angles contains a locally wedge-shaped free boundary minimal hypersurface Σ^n which is smooth in its interior and on its faces and is C^2,α up to and including its edge. We can also handle the case of 90 degree wedge angles under an additional assumption.

4. Generic density of equivariant min-max hypersurfaces

   Tongrui Wang

   arXiv preprint arXiv:2309.09527, 2023

   Abs arXiv

   For a compact Riemannian manifold M^n+1 acted isometrically on by a compact Lie group G with cohomogeneity \rm Cohom(G)≥2, we show the Weyl asymptotic law for the G-equivariant volume spectrum. As an application, we show in the C^\infty_G-generic sense with a certain dimension assumption that the union of min-max minimal G-hypersurfaces (with free boundary) is dense in M, whose boundaries’ union is also dense in \bd M.

5. Improved Hebey-Vaugon conjecture on equivariant Yamabe invariants in dimension 3

   Tongrui Wang, and Xuan Yao

   arXiv preprint arXiv:2309.13861, 2023

   Abs arXiv

   Consider a closed connected 3-manifold M acted diffeomorphically on by a compact Lie group G with at least one orbit of finite cardinality. We show an upper bound for the G-equivariant Yamabe invariant \sigma_G(M) under certain topological assumptions, which improved a conjecture of Hebey-Vaugon.

6. Generalized S^1-stability theorem

   Tongrui Wang, and Xuan Yao

   arXiv preprint arXiv:2309.13865, 2023

   Abs arXiv

   We use the equivariant μ-bubbles technique to prove that for any compact manifold M^n with non-empty boundary, n∈{3,5,6}, the Yamabe invariant of M^n is positive if and only if the Yamabe invariant of M^n\times S^1 is positive. This generalized the S^1-stability conjecture of Rosenberg to compact manifolds with boundary.

Surveys

2023

1. Recent progress on geometric variational theory

   Tongrui Wang, and Xin Zhou

   SCIENTIA SINICA Mathematica, 2023

   Abs HTML

   In this paper, we survey recent progress on the variational theory related to the area functional and the mean curvature. We mainly discuss the Morse theory of minimal hypersurfaces. In particular, we focus on the variational theory for constant mean curvature (CMC) surfaces, the multiplicity one conjecture, generic spatial distributions of minimal hypersurfaces, the variational theory for minimal surfaces with free boundary, and generalizations in the equivariant settings.