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In this paper, we study the Li–Yau's gradient estimate for the p-Laplacian on weighted graphs. For p ≥ 2, under the condition of CDpψ(m, K) curvature, we derive a more general type of Li–Yau's gradient estimate for positive solutions to the p-Laplacian heat equation on finite graphs or locally finite graphs with bounded weighted vertex degree. It is parallel to a result of Kotschwar and Ni on manifolds and an outcome of Wang on smooth metric measure spaces. In particular, when p = 2, our estimate deduces to Lü and Wang's result on graphs with the CDψ(m, K) curvature. As an application of our main result, we derive the corresponding Harnack inequality. [ABSTRACT FROM AUTHOR]

In this paper, we study 3-dimensional complete non-compact Riemannian manifolds with asymptotically nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound. Our main result is that, if this manifold has k ends and finite first Betti number, then it has at most linear volume growth, and furthermore, if the negative part of Ricci curvature decays sufficiently fast at infinity, then we have an optimal asymptotic volume ratio lim sup r → ∞ Vol (B (p , r)) r ≤ 4 k π . In particular, our results apply to 3-dimensional complete non-compact Riemannian manifolds with nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound. [ABSTRACT FROM AUTHOR]

We give a result estimating the dimension of the Lie algebra of Killing vector fields on an irreducible non-trivial gradient Ricci soliton. Then we study the structure of this manifold when the maximal dimension is attained. There are local and global implications. [ABSTRACT FROM AUTHOR]

We are interested in the solution of the Björling problem for timelike surfaces in 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]

We prove that both local and non-local formulations of the Degasperis–Procesi equation possess a pseudospherical nature. As a result, solutions determined by Cauchy problems with non-trivial initial data and a minimal specific regularity define a coframe for a pseudospherical metric within a designated strip. While the region is entirely described by the initial condition, the one-forms are determined by the corresponding solutions. Moreover, we establish that any non-trivial initial condition yields a second fundamental form, which is locally determined within the strip. The surface may collapse within a finite-height strip when the initial condition leads to wave breaking. We determine conditions for the coframe to be defined on the upper plane. We investigate possible integrability structures related to the triad of fundamental one-forms. [ABSTRACT FROM AUTHOR]

We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems: one for the evolution of mean and covariance of the interpolating curve and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities as well as the dynamics of the associated gradient flows. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target. On the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed. [ABSTRACT FROM AUTHOR]

We are interested in the solution of the Björling problem for timelike surfaces in 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]

We prove that on the product of two Riemannian manifolds one of which is compact, any Killing tensor is reducible, that is, is the sum of products of Killing tensors on the factors. The same is true for the lifts to the universal cover of Killing tensors on a compact manifold with reducible holonomy. We give a local description of Killing tensors on product manifolds and present an example of a complete product manifold whose factors are locally irreducible which admits an irreducible Killing tensor field. [ABSTRACT FROM AUTHOR]

We show the existence of a complete, strictly locally convex hypersurface within H that adheres to a curvature equation applicable to a broad range of curvature functions. This hypersurface possesses a prescribed asymptotic boundary at infinity and takes the form of a geodesic graph over a smooth bounded domain Ω at infinity. It is approximated by the shape of geodesic graphs whose boundaries rest upon equidistant hyperplanes. Through this procedure, we establish an alternative method for constructing solutions to the asymptotic Plateau problem comparing to Guan–Spruck–Szapiel–Xiao (J Geom Anal 19(4):772–795, 2009. ; J Differ Geom 96(2):201–222, 2014). [ABSTRACT FROM AUTHOR]

We investigate the topological implications of stable minimal surfaces existing in a static perfect fluid space while ensuring that the fluid satisfies certain energy conditions. Based on the main findings, the topology of the level set { f = c } (the boundary of a stellar model) is studied, where c is a positive constant and f is the static potential of a static perfect fluid space. Bounds for the Hawking mass for the level set { f = c } of a static perfect fluid space are derived. Consequently, we prove an inequality that resembles the Penrose inequality for compact and non-compact static perfect fluid spaces, guaranteeing that the Hawking mass is positive for a class of surfaces in a static perfect fluid space. We will present a section dedicated to examples of static stellar models, one of them inspired by Witten's black hole (or Hamilton's cigar). [ABSTRACT FROM AUTHOR]

On a compact three-dimensional Riemannian manifold with boundary, we prove the compactness of the full set of conformal metrics with positive constant scalar curvature and constant mean curvature on the boundary. This involves a blow-up analysis of a Yamabe equation with critical Sobolev exponents both in the interior and on the boundary. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with noncollapsed Riemannian manifolds (M n , g) with integral curvature bounds, as well as their Gromov–Hausdorff limits (M i n , g i) → GH (X , d) . Our main result generalizes Cheeger's Hausdorff dimension estimate for the singular set in Cheeger (Geom Funct Anal 13:20–72, 2003) and improve it into Minkowski dimension estimate in the spirit of Cheeger and Naber (Invent Math 191:321–339, 2013). We also prove a diffeomorphism finiteness theorem for manifolds in the critical case, using Cheeger–Naber's methods in Cheeger and Naber (Ann Math 1093–1165, 2015). [ABSTRACT FROM AUTHOR]

In this paper, we prove a logarithmic Sobolev inequality for closed submanifolds with constant length of mean curvature vector in a manifold with nonnegative sectional curvature. [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 article, we will study the isoperimetric problem by introducing a mean curvature type flow in the Riemannian manifold endowed with a non-trivial conformal vector field. This flow preserves the volume of the bounded domain enclosed by a star-shaped hypersurface and decreases the area of hypersurface under certain conditions. We will prove the long time existence and convergence of the flow. As a result, the isoperimetric inequality for such a domain is established. Especially, we solve the isoperimetric problem for the star-shaped hypersurfaces in the Riemannian manifold endowed with a closed, non-trivial conformal vector field, a wide class of warped product spaces studied by Guan, Li and Wang is included. [ABSTRACT FROM AUTHOR]

V-static metrics generalise the notion of static metrics, and stem from the work of Miao and Tam (Calc Var Partial Differ Equ 36(2):141–171, 2009), and Corvino, Eichmair, and Miao (Math Ann 357(2):551–584, 2013) on critical points of the volume functional over the space of compact manifolds with constant scalar curvature. In this article we show that these V-static metrics arise naturally in the context of asymptotically hyperbolic manifolds as critical points of the volume-renormalised mass, recently introduced by Dahl, Kröncke, and McCormick (A volume-renormalized mass for asymptotically hyperbolic manifolds. arXiv preprint , 2023). In particular, we show that critical points of the volume-renormalised mass over the space of constant scalar curvature asymptotically hyperbolic manifolds without boundary, or satisfying appropriate boundary conditions, are exactly V-static metrics. This is directly analogous to the relationship between critical points of the ADM mass and static metrics for asymptotically flat manifolds. [ABSTRACT FROM AUTHOR]

This article investigates the geometric and topologic of warped product submanifolds in Riemannian warped product Qεm×R{{\mathbb{Q}}}_{\varepsilon }^{m}\times {\mathbb{R}}. In this respect, we obtain the first Chen inequality that involves extrinsic invariants like the length of the warping functions and the mean curvature. This inequality involves two intrinsic invariants (sectional curvature and δ\delta -invariant). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the Ricci curvature gradient. We aim to apply this theory to many structures and obtain Dirichlet eigenvalues for problem applications. Some new results regarding the vanishing mean curvature are presented as a partial solution, and this can be considered for the well-known problem given by Chern.

In this article, we define Perelman's functionals on manifolds with non-isolated conical singularities by starting from a spectral point of view for the Perelman's λ -functional. (Our definition of non-isolated conical singularities includes isolated conical singularities.) We prove that the spectrum of Schrödinger operator - 4 Δ + R on manifolds with non-isolated conical singularities consists of discrete eigenvalues with finite multiplicities, provided that scalar curvatures of cross sections of cones have a certain lower bound. This enables us to define the λ -functional on these singular manifolds, and further, to prove that the infimum of W-functional is finite, with the help of some weighted Sobolev inequalities. Furthermore, we obtain some asymptotic behavior of eigenfunctions and the minimizer of the W-functional near the singularity, and a more refined optimal partial asymptotic expansion for eigenfunctions near isolated conical singularities. We also study the spectrum of - 4 Δ + R and Perelman's functionals on manifolds with more general singularities, i.e. the r α -horn singularities which serve as prototypes of algebraic singularities. [ABSTRACT FROM AUTHOR]

We derive global curvature estimates for closed, strictly star-shaped (n - 2) -convex hypersurfaces in warped product manifolds, which satisfy the prescribed (n - 2) -curvature equation with a general right-hand side. The proof can be readily adapted to establish curvature estimates for semi-convex solutions to the general k-curvature equation. Furthermore, it can also be used to prove the same estimates for k-convex solutions to the prescribed curvature measure type equations. [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]

Many contact metric manifolds are critical points of curvature functionals restricted to spaces of associated metrics. The Godbillon–Vey functional was never considered in a variational context in Contact Geometry. Recently we extended this functional from foliations to arbitrary plane fields on a three-dimensional manifold, so, the following question arises: can one use the Godbillon–Vey functional to find optimal almost contact manifolds? In the paper, we introduce a Godbillon–Vey type functional for a three-dimensional almost contact manifold, present it in Reinhart–Wood form and find its Euler–Lagrange equations for all variations preserving the Reeb vector field. We construct critical (for our functional) three-dimensional almost contact manifolds having a double-twisted product structure, these solutions belong to the class C 5 ⊕ C 12 according to Chinea-Gonzalez classification. [ABSTRACT FROM AUTHOR]

We study the topology of closed, simply-connected, 6-dimensional Riemannian manifolds of positive sectional curvature which admit isometric actions by SU(2) or SO(3). We show that their Euler characteristic agrees with that of the known examples, i.e., S6, C P 3 , the Wallach space SU(3)/T2 and the biquotient SU(3)//T2. We also classify, up to equivariant diffeomorphism, certain actions without exceptional orbits and show that there are strong restrictions on the exceptional strata. [ABSTRACT FROM AUTHOR]

We provide relationships between the spectral convergences in Bérard–Besson–Gallot sense, in Kasue–Kumura sense and the measured Gromov–Hausdorff convergence, for compact finite dimensional RCD spaces. As an independent interest, a canonical (spectral) approximation map between such spaces constructed by given spectral data is obtained. [ABSTRACT FROM AUTHOR]

We introduce another distinguished closed vector field on a Riemannian manifold which we shall call torse-forming-like and we relate it to other kind of vector fields such as torse-forming, self-torse-forming and semi-torse-forming vector fields. For a warped product manifold, we obtain different types of vector fields on the factor manifolds when the warped product is endowed with a torse-forming-like vector field. We study the nature of the tangential component of such a vector field for some special cases of submanifolds, namely, totally umbilical (in particular, totally geodesic) or having Codazzi (in particular, parallel) shape operator, and we show that the tangential component on a hypersurface of a torse-forming-like vector field is a Killing vector field if and only if the hypersurface is quasi-umbilical. In particular, if the hypersurface is isometrically immersed in a real space-form, then it is a space of quasi-constant curvature, hence, it is a quasi-Einstein manifold. Finally, we characterize the metallic shaped hypersurfaces with the normal direction generated by the normal component of a torse-forming-like vector field having Killing tangential component to the hypersurface. [ABSTRACT FROM AUTHOR]

We study the reach (in the sense of Federer) of the natural isometric embedding X ↪ W p (X) of X inside its p-Wasserstein space, where (X , dist) is a geodesic metric space. We prove that if a point x ∈ X can be joined to another point y ∈ X by two minimizing geodesics, then reach (x , X ⊂ W p (X)) = 0 . This includes the cases where X is a compact manifold or a non-simply connected one. On the other hand, we show that reach (X ⊂ W p (X)) = ∞ when X is a CAT(0) space. The infinite reach enables us to examine the regularity of the projection map. Furthermore, we replicate these findings by considering the isometric embedding X ↪ W ϑ (X) into an Orlicz–Wasserstein space, a generalization by Sturm of the classical Wasserstein space. Lastly, we establish the nullity of the reach for the isometric embedding of X ↪ Dgm ∞ , the space of persistence diagrams equipped with the bottleneck distance. [ABSTRACT FROM AUTHOR]

We consider an aggregation-diffusion energy on Cartan–Hadamard manifolds with sectional curvatures that can grow unbounded at infinity. The energy corresponds to a macroscopic aggregation model that involves nonlocal interactions and linear diffusion. We establish necessary and sufficient conditions on the growth at infinity of the attractive interaction potential for ground states to exist. Specifically, we derive explicit conditions on the attractive potential in terms of the bounds on the sectional curvatures at infinity. To prove our results we establish a new logarithmic Hardy–Littlewood inequality for Cartan–Hadamard manifolds of unbounded curvature. [ABSTRACT FROM AUTHOR]

We consider deformations of the scalar curvature of a partially integrable pseudohermitian manifold, in analogy with the work of Fischer and Marsden on Riemannian manifolds. In particular, we introduce and discuss R-singular spaces, give sufficient conditions for the stability of the scalar curvature, and give a partial infinitesimal rigidity result for the scalar curvature of a compact, torsion-free, scalar-flat, integrable pseudohermitian manifold. [ABSTRACT FROM AUTHOR]

We show that the bordism group of closed 3-manifolds with positive scalar curvature (psc) metrics is trivial by explicit methods. Our constructions are derived from scalar-flat Kähler ALE surfaces discovered by Lock-Viaclovsky. Next, we study psc 4-manifolds with metric singularities along points and embedded circles. Our psc null-bordisms are essential tools in a desingularization process developed by Li-Mantoulidis. This allows us to prove a non-existence result for singular psc metrics on enlargeable 4-manifolds with uniformly Euclidean geometry. As a consequence, we obtain a positive mass theorem for asymptotically flat 4-manifolds with non-negative scalar curvature and low regularity. [ABSTRACT FROM AUTHOR]

Curvature properties of a metric connection with totally skew-symmetric torsion are investigated. It is shown that if either the 3-form T is harmonic, d T = δ T = 0 or the curvature of the torsion connection R ∈ S 2 Λ 2 then the scalar curvature of a ∇ -Einstein manifold is determined by the norm of the torsion up to a constant. It is proved that a compact generalized gradient Ricci soliton with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature are constants. In this case, the torsion 3-form is harmonic and the gradient function has to be constant. Necessary and sufficient conditions for a metric connection with skew torsion to satisfy the Riemannian first Bianchi identity as well as the contracted Riemannian second Binachi identity are presented. It is shown that if the torsion connection satisfies the Riemannian first Bianchi identity then it satisfies the contracted Riemannian second Bianchi identity. It is also proved that a metric connection with skew torsion satisfying the curvature identity R (X , Y , Z , V) = R (Z , Y , X , V) must be flat. [ABSTRACT FROM AUTHOR]

Let Mn be an embedded closed submanifold of ℝk+1 or a smooth bounded domain in ℝn, where n ≥ 3. We show that the local smooth solution to the heat flow of self-induced harmonic map will blow up at a finite time, provided that the initial map u0 is in a suitable nontrivial homotopy class with energy small enough. [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]

We investigate the influence of uniformly positive scalar curvature on the size of a non-collapsed Ricci limit space coming from a sequence of n-manifolds with non-negative Ricci curvature and uniformly positive scalar curvature. We prove that such a limit space splits at most n - 2 lines or R -factors. When this maximal splitting occurs, we obtain a uniform upper bound on the diameter of the non-splitting factor. Moreover, we obtain a volume gap estimate and a volume growth order estimate of geodesic balls on such manifolds. [ABSTRACT FROM AUTHOR]

This paper focuses on a one-dimensional fourth-order nonlinear dispersive partial differential equation for curve flows on a Kähler manifold. The equation arises as a fourth-order extension of the one-dimensional Schrödinger flow equation, with physical and geometrical backgrounds. First, this paper presents a framework that can transform the equation into a system of fourth-order nonlinear dispersive partial differential-integral equations for complex-valued functions. This is achieved by developing the so-called generalized Hasimoto transformation, which enables us to handle general higher-dimensional compact Kähler manifolds. Second, this paper demonstrates the computations to obtain the explicit expression of the derived system for three examples of the compact Kähler manifolds, dealing with the complex Grassmannian as an example in detail. In particular, the result of the computations when the manifold is a Riemann surface or a complex Grassmannian verifies that the expression of the system derived by our framework actually unifies the ones derived previously. Additionally, the computation when the compact Kähler manifold has a constant holomorphic sectional curvature, the setting of which has not been investigated, is also demonstrated. [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]

We propose a generalization of the Minkowski average of two subsets of a Riemannian manifold, in which geodesics are replaced by an arbitrary family of parametrized curves. Under certain assumptions, we characterize families of curves on a Riemannian surface for which a Brunn–Minkowski inequality holds with respect to a given volume form. In particular, we prove that under these assumptions, a family of constant-speed curves on a Riemannian surface satisfies the Brunn–Minkowski inequality with respect to the Riemannian area form if and only if the geodesic curvature of its members is determined by a function κ on the surface, and κ satisfies the inequality K + κ 2 - | ∇ κ | ≥ 0 where K is the Gauss curvature. [ABSTRACT FROM AUTHOR]

We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d > n - 2 2 . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the "singular Yamabe problem". [ABSTRACT FROM AUTHOR]

This short note deals with compact and complete and non-compact gradient Yamabe solitons (M, g, f) such that it has metric of constant scalar curvature. Firstly, we give a new proof of triviality for gradient compact Yamabe solitons. Also, under some integral conditions, we are able to improve a result due to Ma and Miquel (Ann Global Anal Geom 42:195–205, 2012). Finally, we obtain that the Yamabe metric becomes rotationally symmetric. Results for k-Yamabe solitons are also obtained here. [ABSTRACT FROM AUTHOR]

In this paper, we study the locally constrained inverse curvature flow for hypersurfaces in the half-space with θ -capillary boundary, which was recently introduced by Wang et al. (Math Ann 388:2121–2154, 2024). Assume that the initial hypersurface is strictly convex with the contact angle θ ∈ (0 , π / 2 ]. We prove that the solution of the flow remains to be strictly convex for t > 0 , exists for all positive time and converges smoothly to a spherical cap. As an application, we prove a complete family of Alexandrov–Fenchel inequalities for convex capillary hypersurfaces in the half-space with the contact angle θ ∈ (0 , π / 2 ]. Along the proof, we develop a new tensor maximum principle for parabolic equations on compact manifold with proper Neumann boundary condition. [ABSTRACT FROM AUTHOR]