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This paper generalizes D. Burago and S. Ivanov's work (Duke Math J 162:1205–1248, 2013) on filling volume minimality and boundary rigidity of almost real hyperbolic metrics. We show that regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and hence boundary rigid. This includes perturbations of complex, quaternionic and Cayley hyperbolic metrics. [ABSTRACT FROM AUTHOR]

In this paper, (gradient) almost Ricci solitons on Finsler measure spaces (M, F, m) are introduced and investigated. We prove that (M, F, m) is a gradient almost Ricci soliton with soliton scalar κ if and only if the infinity-Ricci curvature Ric ∞ = κ on M. Moreover, we give an equivalent characterization of (gradient) almost Ricci solitons for Randers metrics F = α + β , which implies that every Randers (gradient) almost Ricci soliton is of isotropic S BH -curvature. Based on this and the navigation technique, we further classify Randers almost Ricci solitons (resp., gradient almost Ricci solitons) up to classifications of Randers Einstein metrics F (resp., Riemannian gradient almost Ricci solitons) and the homothetic vector fields of F (resp., solutions of the equation which the weight function f of m satisfies) when F has isotropic S BH -curvature. As applications, we obtain some rigidity results for compact Randers (gradient) Ricci solitons and construct several Randers gradient Ricci solitons, which are the first nontrivial examples of gradient Ricci solitons in Finsler geometry. [ABSTRACT FROM AUTHOR]

In this paper, we establish a Heintze–Karcher-type inequality for capillary hypersurfaces in a unit ball. To achieve this, we introduce a special Finsler metric given by Zermelo's navigation and study the geodesic normal flow with respect to this Finsler metric. Our results indicate that the relationship between capillary hypersufaces and hypersurfaces with free boundary is similar to the one between Finsler geometry and Riemannian geometry. [ABSTRACT FROM AUTHOR]

For a closed minimal immersed hypersurface M in S n + 1 with second fundamental form A, and each integer k ≥ 2 , define a constant σ k = ∫ M A 2 k M . We show that σ k ≥ 2 k provided n = 2 and M is not totally geodesic. When n = 4 and M has two distinct principal curvatures, we show σ 2 ≥ 16 . When n ≥ 3 and M has two distinct principal curvatures, for each integer k ≥ 2 , there exists a positive constant δ k (n) < n , if A 2 ≥ δ k (n) , we have σ k ≥ n k . All the equality holds iff M is isometric to a Clifford hypersurface. [ABSTRACT FROM AUTHOR]

In this paper, we study the rigidity problem for compact minimal Legendrian submanifolds in the unit Euclidean spheres via eigenvalues of fundamental matrices, which measure the squared norms of the second fundamental form on all normal directions. By using Lu's inequality (Lu in J Funct Anal 261:1284–1308, 2011) on the upper bound of the squared norm of Lie brackets of symmetric matrices, we establish an optimal pinching theorem for such submanifolds of all dimensions, giving a new characterization for the Calabi tori. This pinching condition can also be described by the eigenvalues of the Ricci curvature tensor. Moreover, when the third large eigenvalue of the fundamental matrix vanishes everywhere, we get an optimal rigidity theorem under a weaker pinching condition. [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]

In this article, we show super-rigidity of Gromov’s random monster group. It is known from a paper of Assaf Naor and Lior Silberman that any homomorphic image of Gromov’s random monster group into a linear group is finite. It can be also derived from the previously known results that the same result is true for a- L p -menable groups and K-amenable groups. We extend these results and prove that any morphism ϕ α from Gromov’s random monster group Γ α to a countable discrete group G has finite image for almost all α , where G is any of the following types of groups: mapping class group M C G (S g , b) , braid group B n , outer automorphism group of a free group O u t (F N) , automorphism group of a free group A u t (F N) and hierarchically hyperbolic group. For acylindrically hyperbolic groups, we deduce that the homomorphic image is absolutely elliptic. We introduce another property called hereditary super-rigidity, which is the property of super-rigidity for all finite-index sub-groups. It immediately follows from the literature that Γ α has hereditary super-rigidity with respect to an a- L p -menable group or a K-amenable group for a.e. α . In this article, we establish a stability result for the groups with respect to which Γ α has super-rigidity and hereditary super-rigidity. [ABSTRACT FROM AUTHOR]

In this paper, we prove a Poincaré-type inequality for any set of finite perimeter which is stable with respect to the free energy among volume-preserving perturbation, provided that the Hausdorff dimension of its singular set is at most n - 3 . With this inequality, we classify all volume-constraint local energy-minimizing sets in a unit ball, a half-space or a wedge-shaped domain. In particular, we prove that the relative boundary of any energy-minimizing set is smooth. [ABSTRACT FROM AUTHOR]

As an application of the general theory on extrinsic geometry (Doubrov et al. in SIGMA Symmetry Integr Geom Methods Appl 17:061, 2021), we investigate extrinsic geometry in flag varieties and systems of linear PDE’s for a class of special interest associated with the adjoint representation of sl (3) . We carry out a complete local classification of the homogeneous structures in this class. As a result, we find 7 kinds of new systems of linear PDE’s of second order on a 3-dimensional contact manifold each of which has a solution space of dimension 8. Among them there are included a system of PDE’s called contact Cayley’s surface and one which has sl (2) symmetry. [ABSTRACT FROM AUTHOR]

In this paper, we deal with Serrin-type problems in Riemannian manifolds. First, we obtain a Heintze-Karcher inequality and a Soap Bubble result, with its respective rigidity, when the ambient space has a Ricci tensor bounded below. After, we approach a Serrin problem in bounded domains of manifolds endowed with a closed conformal vector field. Our primary tool, in this case, is a new Pohozaev identity, which depends on the scalar curvature of the manifold. Applications involve Einstein and constant scalar curvature spaces. [ABSTRACT FROM AUTHOR]

We obtain some rigidity results for metrics whose Schouten tensor is bounded from below after conformal transformations. Liang Cheng [5] recently proved that a complete, nonflat, locally conformally flat manifold with Ricci pinching condition ( R i c - ϵ R g ≥ 0 ) must be compact. This answers higher dimensional Hamilton's pinching conjecture on locally conformally flat manifolds affirmatively. Since (modified) Schouten tensor being nonnegative is equivalent to a Ricci pinching condition, our main result yields a simple proof of Cheng's theorem. [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]

The classifications of locally strongly convex equiaffine hypersurfaces (resp. centroaffine hypersurfaces) with parallel Fubini–Pick tensor with respect to the Levi-Civita connection of the Blaschke–Berwald affine metric (resp. centroaffine metric) have been completed in the last decades. In this paper we define generalized Calabi products in Calabi geometry and prove decomposition theorems in terms of their Calabi invariants. As the main result, we obtain a complete classification of Calabi hypersurfaces in R n + 1 with parallel Fubini–Pick tensor with respect to the Levi-Civita connection of the Calabi metric. This result is a counterpart in Calabi geometry of the classification theorems in equiaffine situation Hu et al. (J Diff Geom 87:239-307, 2011) and centroaffine situation Cheng et al. (Results Math 72:419-469, 2017). [ABSTRACT FROM AUTHOR]

We show that every local supremum of the systole over the space of Riemannian metrics of curvature at most - 1 on a given nonsimply connected closed surface is attained by a hyperbolic metric. As an application, we also present a partial extension of this result to 3-manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the rigidity of Q-curvature. Specifically, we consider a closed, oriented n-dimensional ( n ≥ 6 ) Riemannian manifold (M, g) and prove the following results under the condition. (1) If (M, g) is locally conformally flat with nonnegative Ricci curvature, then (M, g) is isometric to a quotient of R n , S n , or R × S n - 1 . (2) If (M, g) has δ 2 W = 0 with nonnegative sectional curvature, then (M, g) is isometric to a quotient of the product of Einstein manifolds. Additionally, we investigate some rigidity theorems involving Q-curvature about hypersurfaces in simply-connected space forms. We also show the uniqueness of metrics with constant scalar curvature and constant Q-curvature in a fixed conformal class. [ABSTRACT FROM AUTHOR]

We study the inverse mean curvature flow with a free boundary supported on geodesic spheres in hyperbolic space. Starting from any convex hypersurface inside a geodesic ball with a free boundary, the flow converges to a totally geodesic disk in finite time. Using the convergence result, we show a Willmore type inequality. [ABSTRACT FROM AUTHOR]

In this paper, we consider n-dimensional compact Willmore hypersurface in a unit sphere. We prove a pinching theorem of ρ 2 , which is defined as ρ 2 = S - n H 2 , for n-dimensional compact Willmore hypersurface with constant mean curvature and constant scalar curvature, where H denotes the mean curvature and S the squared norm of the second fundamental form of this hypersurface. [ABSTRACT FROM AUTHOR]

In this paper, we prove a new Heintze–Karcher type inequality for shifted mean convex hypersurfaces in hyperbolic space. As applications, we prove an Alexandrov-type theorem for closed embedded hypersurfaces with constant shifted kth mean curvature in hyperbolic space. Furthermore, two uniqueness results for certain curvature equations are also obtained. [ABSTRACT FROM AUTHOR]

This belongs to a series of papers motivated by Ballmann's Higher Rank Rigidity Conjecture. We prove the following. Let X be a CAT(0) space with a geometric group action Γ ↷ X . Suppose that every geodesic in X lies in an n -flat, n ≥ 2 . If X contains a periodic n -flat which does not bound a flat (n + 1) -half-space, then X is a Riemannian symmetric space, a Euclidean building or non-trivially splits as a metric product. This generalizes the Higher Rank Rigidity Theorem for Hadamard manifolds with geometric group actions. [ABSTRACT FROM AUTHOR]

This paper shows such a sharp Sobolev principle that if (Σ , g) is a compact n-dimensional graphic submanifold of R n + m , G = | det g | is the absolute value of the determinant of g, | B n | is the volume of the open unit ball B n in R n , and f is a positive smooth function on Σ , then ∫ Σ | ∇ f | d V + ∫ ∂ Σ f d A n | B n | 1 n - 1 ≥ ∫ Σ f n n - 1 G - 1 2 d V | B n | 1 n , holds with equality when and only when f = a constant on Σ ; G = 1 on ∂ Σ ; Σ = the image of a round ball in R n under a smooth map. Intrinsically, the foregoing inequality is indeed geometric thanks to the fact that f = G n - 1 2 n = 1 implies the following new sharp isoperimetric inequality for the graphic submanifolds of R n + m with the unit metric determinant | ∂ Σ | n | B n | 1 n - 1 ≥ | Σ | | B n | 1 n. [ABSTRACT FROM AUTHOR]

By carrying out point-wise estimates for the mean curvature, we prove Bernstein type theorems of complete translating solitons of the mean curvature flow in higher codimension under various geometric conditions. [ABSTRACT FROM AUTHOR]

Following Gromov, a Riemannian manifold is called area-extremal if any modification that increases scalar curvature must decrease the area of some tangent 2-plane. We prove that large classes of compact 4-manifolds, with or without boundary, with nonnegative sectional curvature are area-extremal. We also show that all regions of positive sectional curvature on 4-manifolds are locally area-extremal. These results are obtained analyzing sections in the kernel of a twisted Dirac operator constructed from pairs of metrics, and using the Finsler–Thorpe trick for sectional curvature bounds in dimension 4. [ABSTRACT FROM AUTHOR]

In this work, we investigate the geometry and topology of compact Einstein-type manifolds with nonempty boundary. First, we prove a sharp boundary estimate and as consequence we obtain, under certain hypotheses, that the Hawking mass is bounded from below in terms of area. Then we give a topological classification for its boundary. Finally, we deduce some classification results for compact Einstein-type manifolds with positive constant scalar curvature and assuming a pointwise inequality for the traceless Ricci tensor. [ABSTRACT FROM AUTHOR]

In this paper, we investigated the properties of conformal vector fields defined on a Riemannian manifold. Given a conformal vector field X, we can define a skew-symmetric (1, 1)-tensor Φ associated to X which can be used to test whether the 1-form dual to X is closed. First, we show that complete divergence of the associated skew-symmetric (1, 1)-tensor Φ for a conformal vector field X is always vanishing, i.e., div 2 Φ = 0 , and div Φ = 0 if and only if Φ = 0 when M is compact. Second, we consider Riemannian manifolds admitting a conformal vector field whose conformal factor satisfies the critical point equation, and vacuum static spaces admitting a closed conformal vector field whose conformal factor satisfies the vacuum static equation. In both cases, we prove that the given Riemannian manifold is Einstein and is isometric to a standard sphere. These results generalize results in Deshmukh and Alsolamy (Balkan J Geom Appl 17(1):9–16, 2012) and da Silva Filho (Math Nach 293:2299–2305, 2020) in some sense. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce quermassintegrals for capillary hypersurfaces in the half-space. Then we solve the related isoperimetric type problems for the convex capillary hypersurfaces and obtain the corresponding Alexandrov–Fenchel inequalities. In order to prove these results, we construct a new locally constrained curvature flow and prove that the flow converges globally to a spherical cap. [ABSTRACT FROM AUTHOR]

We prove global results about actions of cocompact lattices in higher-rank simple Lie groups on closed manifolds endowed with either a projective class of connections or a conformal class of pseudo-Riemannian metrics of signature (p, q), with min (p , q) ≥ 2 . Building on a recent article [17], provided that such a structure is locally equivalent to its model X , the main question treated here is the completeness of the associated (G , X) -structure. Because of the similarities between the model spaces of projective geometry and non-Lorentzian conformal geometry, a number of arguments apply in both contexts. We therefore present the proofs in parallel. The conclusion is that in both cases, when the real rank is maximal, the manifold is globally equivalent to either the model space X or its double cover. [ABSTRACT FROM AUTHOR]

In some warped product manifolds including space forms, we consider closed self-similar solutions to curvature flows whose speeds are negative powers of mean curvature, Gauss curvature, and other curvature functions with suitable properties. We prove such self-similar solutions, not necessarily strictly convex for some cases, must be slices of warped product manifolds. A new auxiliary function is the key of the proofs. [ABSTRACT FROM AUTHOR]

In this article, the cosymplectic analogue of the symplectic flux homomorphism of a compact connected cosymplectic manifold (M,η,ω)\left(M,\eta ,\omega ) with ∂M=∅\partial M=\varnothing is studied. This is a continuous map with respect to the C0{C}^{0}-metric, whose kernel is connected by smooth arcs and coincides with the subgroup of all weakly Hamiltonian diffeomorphisms. We discuss the cosymplectic analogue of the Weinstein’s chart, and derive that the group Gη,ω(M){G}_{\eta ,\omega }\left(M) of all cosymplectic diffeomorphisms isotopic to the identity map is locally contractible. A study of an analogue of Polterovich’s regularization process for co-Hamiltonian isotopies follows. Finally, we study Moser’s stability theorems for locally conformal cosymplectic manifolds.

We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to $L^p$ -perturbations of their fundamental forms: For a manifold ${\mathcal M}^d$ endowed with a reference metric and a reference shape operator, we show that a sequence of immersions $f_n:{\mathcal M}^d\to {\mathcal N}^{d+1}$ , whose pullback metrics and shape operators are arbitrary close in $L^p$ to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result [AKM22] to a general target manifold ${\mathcal N}$ , removing a constant curvature assumption. The method of proof differs from that in [AKM22]: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to [CMM19] but with no a priori assumed bounds.

In this paper, we give a local rigid characterization of all homogeneous holomorphic two-spheres in a complex Grassmann manifold G(2, N). Let κ denote the new global invariant defined in terms of the square norm of (1, 0) part of the second-order covariant differential of the first ∂ -transform for a holomorphic curve in G(2, N). It is proved that a linearly full non-degenerate holomorphic curve of constant curvature and constant square norm of the second fundamental form in G(2, N) with κ = 0 must be homogeneous. [ABSTRACT FROM AUTHOR]

In this paper, we study gradient Yamabe solitons realized as warped product manifolds. We apply the maximum principle to find lower bound estimates for both the potential function of the soliton and the scalar curvature of the warped product manifold. By slightly modifying Li–Yau's technique, we can handle the drifted Laplacian and thus find different gradient estimates for the warping function according to the sign of the (constant) scalar curvature of the fiber manifold. We close the article with a theorem stating the nonexistence of gradient Yamabe solitons on top of warped products with a base manifold satisfying certain analytical hypothesis. [ABSTRACT FROM AUTHOR]

We show that a singular Riemannian foliation of codimension two on a compact simply-connected Riemannian (n + 2) -manifold, with regular leaves homeomorphic to the n-torus, is given by a smooth effective n-torus action. This solves in the negative for the codimension 2 case a question about the existence of foliations by exotic tori on simply-connected manifolds. [ABSTRACT FROM AUTHOR]

Zhang (2021), Luo and Yin (2022) initiated the study of Lagrangian submanifolds satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂn or ℂℙn, where T = ∇ ∗ h ˜ and h ˜ is the Lagrangian trace-free second fundamental form. They proved several rigidity theorems for Lagrangian surfaces satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂ2 under proper small energy assumption and gave new characterization of the Whitney spheres in ℂ2. In this paper, the authors extend these results to Lagrangian submanifolds in ℂn of dimension n ≥ 3 and to Lagrangian submanifolds in ℂℙn. [ABSTRACT FROM AUTHOR]

Zhang (2021), Luo and Yin (2022) initiated the study of Lagrangian submanifolds satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂn or ℂℙn, where T = ∇ ∗ h ˜ and h ˜ is the Lagrangian trace-free second fundamental form. They proved several rigidity theorems for Lagrangian surfaces satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂ2 under proper small energy assumption and gave new characterization of the Whitney spheres in ℂ2. In this paper, the authors extend these results to Lagrangian submanifolds in ℂn of dimension n ≥ 3 and to Lagrangian submanifolds in ℂℙn. [ABSTRACT FROM AUTHOR]

In this paper, we establish a Heintze–Karcher-type inequality for hypersurfaces with capillary boundary of contact angle θ ∈ (0 , π 2 ] in the half-space or a half ball, by using solution to a mixed boundary value problem in Reilly type formula. Consequently, we give a new proof of Alexandrov-type theorem for embedded capillary constant mean curvature hypersurfaces with contact angle θ ∈ (0 , π 2 ] in the half-space or a half ball. [ABSTRACT FROM AUTHOR]

Abstract The purpose of the present paper is to examine the isometries of almost Ricci–Yamabe solitons. Firstly, the conditions under which a compact gradient almost Ricci–Yamabe soliton is isometric to Euclidean sphere $$S^n(r)$$ S n ( r ) are obtained. Moreover, we have shown that the potential f of a compact gradient almost Ricci–Yamabe soliton agrees with the Hodge–de Rham potential h. Next, we studied complete gradient almost Ricci–Yamabe soliton with $$\alpha \ne 0$$ α ≠ 0 and non-trivial conformal vector field with non-negative scalar curvature and proved that it is either isometric to Euclidean space $$E^n$$ E n or Euclidean sphere $$S^n.$$ S n . Also, solenoidal and torse-forming vector fields are considered. Lastly, some non-trivial examples are constructed to verify the obtained results.

The purpose of the present paper is to examine the isometries of almost Ricci–Yamabe solitons. Firstly, the conditions under which a compact gradient almost Ricci–Yamabe soliton is isometric to Euclidean sphere S n (r) are obtained. Moreover, we have shown that the potential f of a compact gradient almost Ricci–Yamabe soliton agrees with the Hodge–de Rham potential h. Next, we studied complete gradient almost Ricci–Yamabe soliton with α ≠ 0 and non-trivial conformal vector field with non-negative scalar curvature and proved that it is either isometric to Euclidean space E n or Euclidean sphere S n. Also, solenoidal and torse-forming vector fields are considered. Lastly, some non-trivial examples are constructed to verify the obtained results. [ABSTRACT FROM AUTHOR]

In this note, we show that compact static near-horizon geometries with negative cosmological constant are either Einstein or the product of a circle and an Einstein metric. Chruściel, Reall, and Todd proved rigidity when the cosmological constant vanishes, in which case one get the stronger result that the space is Ricci flat (Chruściel et al. in Class Quantum Gravity 23:549–554, 2006). It has been previously asserted that a stronger rigidity statement also holds for negative cosmological constant, but Bahuaud, Gunasekaran, Kunduri, and Woolgar recently pointed out that this was not the case (Bahuaud et al. in Lett Math Phys 112(6):116, 2022). They showed, moreover, that for a compact static near-horizon geometry with negative cosmological constant, the potential vector field X is constant length and divergence-free. We give an argument using the Bochner formula to improve their conclusion to X being a parallel field, which implies the optimal rigidity result. The result also holds more generally for m-Quasi Einstein metrics with m > 0 . [ABSTRACT FROM AUTHOR]

We show that the discrete approximate volume preserving mean curvature flow in the flat torus T N starting near a strictly stable critical set E of the perimeter converges in the long time to a translate of E exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided. [ABSTRACT FROM AUTHOR]