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We prove that in any rank one symmetric space of non-compact type M ∈ { ℝ ⁢ H n , ℂ ⁢ H m , ℍ ⁢ H m , 핆 ⁢ H 2 } {M\in\{\mathbb{R}H^{n},\mathbb{C}H^{m},\mathbb{H}H^{m},\mathbb{O}H^{2}\}} , geodesic spheres are uniformly quantitatively stable with respect to small C 1 {C^{1}} -volume preserving perturbations. We quantify the gain of perimeter in terms of the W 1 , 2 {W^{1,2}} -norm of the perturbation, taking advantage of the explicit spectral gap of the Laplacian on geodesic spheres in M. As a consequence, we give a quantitative proof that for small volumes, geodesic spheres are isoperimetric regions among all sets of finite perimeter. [ABSTRACT FROM AUTHOR]

Let S 1 , S 2 , S 3 {S_{1},S_{2},S_{3}} be oriented non-flat minimal surfaces immersed in ℝ n + 1 {{\mathbb{R}}^{n+1}} ( n ≥ 2 ) {(n\geq 2)} with the Gauss maps G 1 , G 2 , G 3 {G_{1},G_{2},G_{3}} into ℙ n ⁢ ( ℂ ) {{\mathbb{P}}^{n}({\mathbb{C}})} , respectively. Assume that there are conformal diffeomorphisms Φ 2 , Φ 3 {\Phi_{2},\Phi_{3}} of S 1 {S_{1}} onto S 2 , S 3 {S_{2},S_{3}} respectively and let f 1 = G 1 {f^{1}=G_{1}} , f 2 = G 2 ∘ Φ 2 {f^{2}=G_{2}\circ\Phi_{2}} , f 3 = G 3 ∘ Φ 3 {f^{3}=G_{3}\circ\Phi_{3}} . In this paper, we will show that f 1 , f 2 , f 3 {f^{1},f^{2},f^{3}} are algebraic dependence, i.e., f 1 ∧ f 2 ∧ f 3 ≡ 0 {f^{1}\wedge f^{2}\wedge f^{3}\equiv 0} , if they have the same inverse images for a few hyperplanes of ℙ n ⁢ ( ℂ ) {{\mathbb{P}}^{n}({\mathbb{C}})} in general position with some certain conditions. [ABSTRACT FROM AUTHOR]

In this paper, we establish a Simons-type integral inequality for minimal surfaces with constant Kähler angle in complex projective spaces, and we determine all the closed minimal surfaces with the square norm of the second fundamental form satisfying a pinching condition. [ABSTRACT FROM AUTHOR]

In this paper, we study the geometric classifications of λ -self-expanders in three-dimensional Euclidean space. It is proved that for a 2-dimensional λ -self-expander X : M → R 3 in the 3-dimensional Euclidean space R 3 , if the Gauss curvature K is constant, then locally M is a plane R 2 , a cylinder S 1 ( λ ± λ 2 - 4 2) × R , a round sphere S 2 ( λ ± λ 2 - 8 4) and a generalized cylinder Γ × R , where Γ is an λ -self-expander curve. Furthermore, we give a classification of λ -self-expander M with constant squared norm of the second fundamental form, and show that M is locally a plane R 2 , a cylinder S 1 ( λ ± λ 2 - 4 2) × R and a round sphere S 2 ( λ ± λ 2 - 8 4) . [ABSTRACT FROM AUTHOR]

We deal with several aspects of the geometry of m-dimensional mean curvature flow solitons immersed in a Riemannian warped product I × f M n ( m ≤ n ), with base I ⊂ R , fiber M n and warping function f ∈ C ∞ (I) . In this context, we apply suitable maximum principles to guarantee that such a mean curvature flow soliton is a slice of the ambient space, as well as to obtain nonexistence results concerning these geometric objects. When m = n , we investigate complete two-sided hypersurfaces and, in particular, entire graphs constructed over the fiber M n which are mean curvature flow solitons. Furthermore, we infer the stability of closed mean curvature flow solitons with respect to an appropriate stability operator. Applications to self-shrinkers and self-expanders in the Euclidean space and to mean curvature flow solitons in important ambient spaces, like the pseudo-hyperbolic, Schwarzschild and Reissner–Nordström spaces, are also given. [ABSTRACT FROM AUTHOR]

In this paper, we study minimal Legendrian surfaces Σ immersed in the tangent sphere bundle T1ℝ3. We classify (1) totally geodesic Legendrian surfaces, (2) closed minimal Legendrian surfaces of genus smaller than or equal to 1 and complete minimal Legendrian surfaces with the non-negative Gauss curvature, and (3) complete stable minimal Legendrian surfaces. [ABSTRACT FROM AUTHOR]

We use an intrinsic Klotz–Osserman type result for surfaces in terms of Codazzi operators to study surfaces with parallel mean curvature and non-positive Gaussian curvature in product spaces. [ABSTRACT FROM AUTHOR]

We study strong half-space theorems for the classes of complete 1-surfaces with bounded curvature, parabolic 1-surfaces, and stochastically complete H-surfaces with H < 1 immersed in the hyperbolic space H 3 . As a by-product of the techniques we obtain a Maximum Principle at Infinity for 1-surfaces in H 3 . We also address the intersection problem for 1-surfaces immersed in a complete Riemannian three-manifold P with Ricci curvature bounded from below by - 2 . We establish a splitting result provided the distance between the 1-surfaces is realized and Ric P ≥ - 2 , and a Frankel’s type theorem for 1-surfaces with bounded curvature immersed in P when Ric P > - 2 . [ABSTRACT FROM AUTHOR]

In this paper we study isometric immersions f : M n → C ′ P n of an n-dimensional pseudo-Riemannian manifold M n into the n-dimensional para-complex projective space C ′ P n . We study the immersion f by means of a lift f of f into a quadric hypersurface in S n + 1 2 n + 1 . We find the frame equations and compatibility conditions. We specialize these results to dimension n = 2 and a definite metric on M 2 in isothermal coordinates and consider the special cases of Lagrangian surface immersions and minimal surface immersions. We characterize surface immersions with special properties in terms of primitive harmonicity of the Gauss maps. [ABSTRACT FROM AUTHOR]

Given a function H ∈ C 1 (S 2) , an H -surface Σ is a surface in the Euclidean space R 3 whose mean curvature H Σ satisfies H Σ = H ∘ η , where η is the Gauss map of Σ . The purpose of this paper is to use a phase space analysis to give some classification results for helicoidal H -surfaces, when H is rotationally symmetric, that is, H ∘ η = h ∘ ν , for some h ∈ C 1 ([ - 1 , 1 ]) , where ν is the angle function of the surface. We prove a classification theorem for the case where h (t) is even and increasing for t ∈ [ 0 , 1 ] . Finally, we provide examples of helicoidal H -surfaces in cases where h vanishes at some point. [ABSTRACT FROM AUTHOR]

We study complete minimal surfaces in R n with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy W : = 1 4 ∫ | H → | 2 . In codimension one, we prove that the W -Morse index for any inverted minimal sphere or real projective plane with m such ends is exactly m - 3 = W 4 π - 3 . We also consider several geometric properties—for example, the property that all m asymptotic planes meet at a single point—of these minimal surfaces and explore their relation to the W -Morse index of their inverted surfaces. [ABSTRACT FROM AUTHOR]

In this paper I will revisit the construction of a global weak solution to the volume preserving mean curvature flow via discrete minimizing movement scheme by Mugnai, Seis and Spadaro [L. Mugnai, C. Seis and E. Spadaro, Global solutions to the volume-preserving mean-curvature flow, Calc. Var. Partial Differential Equations 55 2016, 1, Article ID 18]. This method is based on the gradient flow approach due to Almgren, Taylor and Wang [F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim. 31 1993, 2, 387–438] and Luckhaus and Sturzenhecker [S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations 3 1995, 2, 253–271] and my aim is to replace the volume penalization with the volume constraint directly in the discrete scheme, which from practical point of view is perhaps more natural. A technical novelty is the proof of the density estimate which is based on second variation argument. [ABSTRACT FROM AUTHOR]

Given ε 0 > 0 , I ∈ ℕ ∪ { 0 } and K 0 , H 0 ≥ 0 , let X be a complete Riemannian 3-manifold with injectivity radius Inj ⁡ (X) ≥ ε 0 and with the supremum of absolute sectional curvature at most K 0 , and let M ↬ X be a complete immersed surface of constant mean curvature H ∈ [ 0 , H 0 ] with index at most I. For such M ↬ X , we prove a structure theorem which describes how the interesting ambient geometry of the immersion is organized locally around at most I points of M, where the norm of the second fundamental form takes on large local maximum values. [ABSTRACT FROM AUTHOR]

The Clifford 2-torus in S 3 and the equlaterial 2-torus in S 5 are known as the only minimal immersions of 2-tori into spheres by the first eigenfunctions (called λ 1 -minimal for short). For n ≥ 3 , the Clifford n-torus in S 2 n - 1 might be the only known example of λ 1 -minimal n-tori in the literature. By discussing the general construction of homogeneous minimal flat n-tori in spheres, we construct several new examples of λ 1 -minimal flat 3-tori and 4-tori. In particular, the existence of 2-parameter family of non-congruent λ 1 -minimal flat 4-tori is shown for the first time. We obtain the complete classification for λ 1 -minimal immersions of conformally flat 3-tori and 4-tori in spheres, by some detailed investigations of shortest vectors in lattices, which could be of independent interests. Using them, we also solve the Berger's problem (finding the maximal value of the dilation-invariant functional λ 1 (g) V (g) 2 n ) among all flat 3-tori and 4-tori. [ABSTRACT FROM AUTHOR]

In this paper, we classify the non-degenerate ruled surfaces which are homothetic α -self-similar solutions to the mean curvature flow (MCF) in Minkowski 3-space E 1 3 . Other than cylindrical surfaces as in E 3 , we find a class of non-cylindrical ruled surface with null rulings, in particular, the null scrolls. Further, we investigate the non-degenerate surfaces of revolutions generated by non-null curve as homothetic α -self-similar solutions to the MCF according to the causality of their rotation axes as spacelike, timelike and lightlike. [ABSTRACT FROM AUTHOR]

L ν operator is an important extrinsic differential operator of divergence type and has profound geometric settings. In this paper, we consider the clamped plate problem of L ν 2 operator on a bounded domain of the complete Riemannian manifolds. A general formula of eigenvalues of L ν 2 operator is established. Applying this general formula, we obtain some estimates for the eigenvalues with higher order on the complete Riemannian manifolds. As several fascinating applications, we discuss this eigenvalue problem on the complete translating solitons, minimal submanifolds on the Euclidean space, submanifolds on the unit sphere and projective spaces. In particular, we get a universal inequality with respect to the L I I operator on the translating solitons. Usually, it is very difficult to get universal inequalities for weighted Laplacian and even Laplacian on the complete Riemannian manifolds. Therefore, this work can be viewed as a new contribution to universal estimate. [ABSTRACT FROM AUTHOR]

In this paper we study the stability of a Killing cylinder in hyperbolic 3-space when regarded as a capillary surface for the partitioning problem. In contrast with the Euclidean case, we consider a variety of totally umbilical support surfaces, including horospheres, totally geodesic planes, equidistant surfaces and round spheres. In all of them, we explicitly compute the Morse index of the corresponding eigenvalue problem for the Jacobi operator. We also address the stability of compact pieces of Killing cylinders with Dirichlet boundary conditions when the boundary is formed by two fixed circles, exhibiting an analogous to the Plateau–Rayleigh instability criterion for Killing cylinders in the Euclidean space. Finally, we prove that the Delaunay surfaces can be obtained by bifurcating Killing cylinders supported on geodesic planes. [ABSTRACT FROM AUTHOR]

We demonstrate that the property of being Alexandrov immersed is preserved along mean curvature flow. Furthermore, we demonstrate that mean curvature flow techniques for mean convex embedded flows such as noncollapsing and gradient estimates also hold in this setting. We also indicate the necessary modifications to the work of Brendle–Huisken to allow for mean curvature flow with surgery in the Alexandrov immersed, 2-dimensional setting. [ABSTRACT FROM AUTHOR]

We prove an abstract structure theorem for weighted manifolds supporting a weighted f-Poincaré inequality and whose ends satisfy a suitable non-integrability condition. We then study how our arguments can be used to obtain full topological control on two important classes of hypersurfaces of the Euclidean space, namely translators and self-expanders for the mean curvature flow, under either stability or curvature asumptions. As an important intermediate step in order to get our results we get the validity of a Poincaré inequality with respect to the natural weighted measure on any translator and we prove that any end of a translator must have infinite weighted volume. Similar tools can be obtained for properly immersed self-expanders permitting to get topological rigidity under curvature assumptions. [ABSTRACT FROM AUTHOR]

Let Γ be a smooth, closed, oriented, (n - 1) -dimensional submanifold of R n + 1 . It was shown by Chodosh–Mantoulidis–Schulze [6] that one can perturb Γ to a nearby Γ ′ such that all minimizing currents with boundary Γ ′ are smooth away from a set with Hausdorff dimension less than n - 9 . We prove that the perturbation can be made such that the singular set of the minimizing current with boundary Γ ′ has Minkowski dimension less than n - 9 . [ABSTRACT FROM AUTHOR]

Our purpose is to establish nonexistence results concerning complete noncompact mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products, under mild constraints on the warping and soliton functions. Applications to self-shrinkers in the Euclidean space, as well as to mean curvature flow solitons in other important warped product models such as the Schwarzschild and Reissner-Nordström spaces, and Robertson-Walker spacetimes such as the Einstein-de Sitter spacetime, are also given. Furthermore, we study the nonexistence of entire solutions to the mean curvature flow equation. Our approach is based on a suitable conformal change of metric jointly with a maximum principle for complete noncompact Riemannian manifolds with polynomial volume growth due to Alías et al.

In this paper, we study biharmonic Riemannian submersions π : M 2 × R → (N 2 , h) from a product manifold onto a surface and obtain some local characterizations of such biharmonic maps. Our results show that when the target surface is flat, a proper biharmonic Riemannian submersion π : M 2 × R → (N 2 , h) is locally a projection of a special twisted product, and when the target surface is non-flat, π is locally a special map between two warped product spaces with a warping function that solves a single ODE. As a by-product, we also prove that there is a unique proper biharmonic Riemannian submersion H 2 × R → R 2 given by the projection of a warped product onto the Euclidean plane. [ABSTRACT FROM AUTHOR]

In this paper, we study the universal inequalities for eigenvalues of a clamped plate problem, and establish some new universal inequalities that are different from those already present in the literature, such as (Wang and Xia J. Funct. Anal. 245(1), 334-352 2007), (Wang and Xia Calc. Var. Partial Differential 653 Equations 40(1-2), 273-289 2011), (Chen, Zheng, and Lu Pacific J. Math. 255(1), 41-54 2012), and so on. In particular, our results can reveal the relationship between the (k + 1) -th eigenvalue and the first k eigenvalues relatively quickly. [ABSTRACT FROM AUTHOR]

We prove instability of Type (II) Lawson–Osserman cones in Euclidean spaces, and thus provide a family of (uncountably many) non-smooth unstable minimal graphs of high codimension versus smooth unstable ones by Lawson–Osserman using a min–max technique. To our knowledge, these are the first examples of non-smooth unstable minimal graphs. [ABSTRACT FROM AUTHOR]

We propose a notion of stability for capillary hypersurfaces with constant higher order mean curvature and we generalize some results of the classical stability theory for constant mean curvature capillary hypersurfaces. [ABSTRACT FROM AUTHOR]

In this paper, we first derive a quantitative quermassintegral inequality for nearly spherical sets in H n + 1 and S n + 1 , which is a generalization of the quantitative Alexandrov–Fenchel inequality proved in R n + 1 (VanBlargan and Wang, Commun Contemp Math 26(06):2350026, 2024). Then we use this method to derive the stability of some geometric inequalities involving weighted curvature integrals and quermassintegrals for nearly spherical sets in R n + 1 and H n + 1 . [ABSTRACT FROM AUTHOR]

In this paper, we estimate areas of compact minimal rotational hypersurfaces in the unit sphere. A sharper lower bound and upper bound on areas of these hypersurfaces are given. As applications, we confirm two conjectures, one in Perdomo and Wei (Nonlinear Anal 125:241–250, 2015), another in Cheng et al (Asian J Math 25:183–194, 2021) and give estimations of entropies of some special self-shrinkers. [ABSTRACT FROM AUTHOR]

We show that, for any given q ≥ 0 {q\geq 0} , any Sasakian structure on a closed manifold M is approximated in the C q {C^{{q}}} -norm by structures induced by CR embeddings into weighted Sasakian spheres. In order to obtain this result, we also strengthen the approximation of an orbifold Kähler form by projectively induced ones given in [J. Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom. 88 2011, 1, 109–159] in the C 0 {C^{0}} -norm to a C q {C^{{q}}} -approximation. [ABSTRACT FROM AUTHOR]

A theorem by Almgren establishes that any minimal 2-sphere immersed in S 3 is a totally geodesic equator. In this paper we give a purely geometric extension of Almgren's result, by showing that any immersed, real analytic 2-sphere in S 3 that is saddle, i.e., of non-positive extrinsic curvature, must be an equator of S 3 . We remark that, contrary to Almgren's theorem, no geometric PDE is imposed on the surface. The result is not true for C ∞ spheres. [ABSTRACT FROM AUTHOR]

We consider smooth Riemannian surfaces whose curvature K satisfies the relation Δ log | K - c | = a K + b away from points where K = c for some (a , b , c) ∈ R 3 , which we call generalized Ricci surfaces. We prove some isometric immersion theorems allowing points where K = c using properties of log-harmonic functions. For instance, we obtain a characterization of Riemannian surfaces that locally admit minimal isometric immersions, possibly with umbilical points, into a 3-dimensional Riemannian manifold of constant sectional curvature. We also give an application to convex affine spheres. Finally, we study compact generalized Ricci surfaces: we obtain topological obstructions and construct examples. [ABSTRACT FROM AUTHOR]

We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and provide a number of geometric applications. In particular, we derive an inequality which relates the eigenvalues of the Jacobi operator for f-minimal hypersurfaces and the spectrum of the Hodge Laplacian. [ABSTRACT FROM AUTHOR]

In the recent paper (Wang et al. in Differ Geom Appl 80:101840, 2022), the authors and Xu have established a Simons-type integral inequality for holomorphic curves in a complex Grassmann manifold G(k, N). In this paper, we completely classify holomorphic immersions from the two-sphere of constant curvature into G(3, N) with the norm of the second fundamental form satisfying the equality case of the inequality and prove that any such immersion can be decomposed as the “direct sum” of some “foundation stones” up to congruence. [ABSTRACT FROM AUTHOR]

We prove that on every compact Riemann surface M, there is a Cantor set C ⊂ M such that M \ C admits a proper conformal constant mean curvature one (CMC-1 ) immersion into hyperbolic 3-space H 3 . Moreover, we obtain that every bordered Riemann surface admits an almost proper CMC-1 face into de Sitter 3-space S 1 3 , and we show that on every compact Riemann surface M, there is a Cantor set C ⊂ M such that M \ C admits an almost proper CMC-1 face into S 1 3 . These results follow from different uniform approximation theorems for holomorphic null curves in C 2 × C ∗ that we also establish in this paper. [ABSTRACT FROM AUTHOR]

Motivated by a conjecture in Loi et al. (Math Zeit 290:599–613, 2018) we prove that the Kähler cone over a regular complete Sasakian manifold is Ricci-flat and projectively induced if and only if it is flat. We also obtain that, up to D a —homothetic transformations, Kähler cones over homogeneous compact Sasakian manifolds are projectively induced. As main tool we provide a relation between the Kähler potentials of the transverse Kähler metric and of the cone metric. [ABSTRACT FROM AUTHOR]

This approach finds new upper bounds for the first positive eigenvalue of the p-Laplacian operator using the mean and constant sectional curvatures on Riemannian manifolds. In particular, we provide several estimates for the first nonzero eigenvalue of the p-Laplacian operator on closed orientated totally real submanifolds of dimension m in a generalized complex space form M n (κ , ε) . Moreover, we generalize the Reilly-inequality of Laplacian (Reilly in Comment Math Helv 52(4):525–533, 1977) to the p-Laplacian for totally real submanifold in complex projective space and complex Euclidean space for κ = 1 and κ = 0 , as applications, respectively. [ABSTRACT FROM AUTHOR]

On the two-sphere Σ , we consider the problem of minimising among suitable immersions f : Σ → R 3 the weighted L ∞ norm of the mean curvature H, with weighting given by a prescribed ambient function ξ , subject to a fixed surface area constraint. We show that, under a low-energy assumption which prevents topological issues from arising, solutions of this problem and also a more general set of "pseudo-minimiser" surfaces must satisfy a second-order PDE system obtained as the limit as p → ∞ of the Euler–Lagrange equations for the approximating L p problems. This system gives some information about the geometric behaviour of the surfaces, and in particular implies that their mean curvature takes on at most three values: H ∈ { ± ‖ ξ H ‖ L ∞ } away from the nodal set of the PDE system, and H = 0 on the nodal set (if it is non-empty). [ABSTRACT FROM AUTHOR]

We study immersions of Sasakian manifolds into finite and infinite dimensional Sasakian space forms. After proving Calabi’s rigidity results in the Sasakian setting, we characterise all homogeneous Sasakian manifolds which admit a (local) Sasakian immersion into a nonelliptic Sasakian space form. Moreover, we give a characterisation of homogeneous Sasakian manifolds which can be embedded into the standard sphere both in the compact and noncompact case. [ABSTRACT FROM AUTHOR]

In this paper, we study the Dirichlet problem for p-convex hypersurfaces with prescribed curvature. We prove that there exists a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition. An interior curvature estimate is also obtained. [ABSTRACT FROM AUTHOR]

We are interested in the solution of the Björling problem for timelike surfaces in R 1 4 \mathbb{R}_{1}^{4} . The main contribution of the paper is to present new and many examples of timelike zero mean curvature surfaces and give their explicit parametric equations. In particular cases, one observes that the parametric equations of these surfaces coincide with the timelike minimal surfaces in Lorentz–Minkowski 3-space. [ABSTRACT FROM AUTHOR]