Ricci Curvature Research Articles

Global Ricci curvature behaviour for the Kähler-Ricci flow with finite time singularities

Three-manifolds with non-negatively pinched Ricci curvature

Abstract Choi-Wang obtained a lower bound of the first eigenvalue of the Laplacian on closed minimal hypersurfaces. On minimal hypersurfaces with boundary, Fraser-Li established an inequality giving a lower bound of the first Steklov eigenvalue as a counterpart of the Choi-Wang inequality. The Fraser-Li type inequality was obtained for manifolds with non-negative Ricci curvature. In this paper, we extend it to the setting of non-negative Ricci curvature with respect to the Wylie-Yeroshkin type affine connection. Our results apply to both weighted Riemannian manifolds with non-negative 1-weighted Ricci curvature and substatic triples.

Positive intermediate Ricci curvature on connected sums

Abstract The classical Minkowski inequality implies that the volume of a bounded convex domain in the Euclidean space is controlled from above by the integral of the mean curvature of its boundary. In this note, an analogous inequality is established without assuming convexity, valid for all bounded smooth domains in a complete manifold whose bottom spectrum is suitably large relative to its Ricci curvature lower bound. An immediate consequence is the nonexistence of embedded closed minimal hypersurfaces in such manifolds. The same nonexistence issue is also addressed for steady and expanding Ricci solitons. The proofs are very much inspired by a sharp monotonicity formula, derived for positive harmonic functions on manifolds with positive spectrum.

Abstract In this paper, we derive a black hole solution within the Einstein–Maxwell framework incorporating a non-minimal coupling between the Ricci tensor and the Maxwell field strength tensor, using a perturbative approach. We subsequently explore the thermodynamic phase transitions of the black hole in an extended phase space, analyzing both canonical and grand canonical ensembles. Our findings reveal that the system exhibits Van der Waals–like behavior in both ensembles. Moreover, for sufficiently small values of electric charge and Maxwell potential, the thermodynamics is dominated by a Hawking–Page phase transition.

Internal spacetime geometry was recently introduced to model certain quantum phenomena using spacetime metrics that are degenerate. We use the Ricci tensors of these metrics to derive a ratio of the bare up and down quark masses, obtaining [Formula: see text]. This value is within the lattice QCD value [Formula: see text], obtained at [Formula: see text] in the minimal subtraction scheme using supercomputers. Moreover, using the Levi-Civita Poisson equation, we derive ratios of the dressed electron mass and bare quark masses. For a dressed electron mass of [Formula: see text], these ratios yield the bare quark masses [Formula: see text] and [Formula: see text], which are within/near the lattice QCD values [Formula: see text] and [Formula: see text]. Finally, using [Formula: see text]-accelerations, we derive the ratio [Formula: see text] of the constituent up and down quark masses. This value is within the [Formula: see text] range of constituent quark models. All of the ratios we obtain are from first principles alone, with no free or ad hoc parameters. Furthermore, and rather curiously, our derivations do not use quantum field theory, but only tools from general relativity.

The prescribed Ricci curvature problem consists of finding a Riemannian metric $g$ to satisfy the equation $Ric(g) = T$, for some fixed symmetric $(0,2)$-tensor field $T$ on a differential manifold $M$. In this paper, we define Schouten-like metric as a particular solution of a prescribed Ricci curvature problem, and we classify them on five-dimensional nilpotent Lie groups by establishing a link with algebraic Schouten solitons.

We consider solutions to some semilinear elliptic equations on complete noncompact Riemannian manifolds and study their classification as well as the effect of their presence on the underlying manifold. When the Ricci curvature is nonnegative, we prove both the classification of positive solutions to the critical equation and the rigidity for the ambient manifold. The same results are obtained when we consider solutions to the Liouville equation on Riemannian surfaces. The results are obtained via a suitable P -function whose constancy implies the classification of both the solutions and the underlying manifold. The analysis carried out on the P -function also makes it possible to classify nonnegative solutions for subcritical equations on manifolds enjoying a Sobolev inequality and satisfying an integrability condition on the negative part of the Ricci curvature. Some of our results are new even in the Euclidean case.

In this paper, we study Z-solitons and gradient Z-solitons on α-cosymplectic manifolds. The soliton structure is defined by the generalized tensor Z=S+βg, where S denotes the Ricci tensor, g the metric tensor, and β a smooth function. We investigate the geometric implications of Z-solitons under various curvature conditions, with a focus on the interplay between the Z-tensor and the Q-curvature tensor, as well as the case of Z-recurrent α-cosymplectic manifolds. Our classification results establish that such manifolds can be Einstein, η-Einstein, or of constant curvature. Finally, we construct a concrete five-dimensional example of an α-cosymplectic manifold that admits a Z-soliton structure, thereby illustrating the theoretical framework.

In this paper, we introduce and investigate a novel space-time, named a pseudo generalized Z-recurrent spacetime. Next, we illustrate that a pseudo generalized Z-recurrent spacetime with Codazzi type of Ricci tensor is a pseudo quasi-Einstein spacetime under certain condition; whereas the space-time is reduced to a nearly quasi-Einstein spacetime if we consider a parallel Ricci tensor. Moreover, we establish that a pseudo generalized Z-recurrent generalized Robertson–Walker space-time becomes a perfect fluid spacetime. Finally, we examine the effect of such spacetime under [Formula: see text] and [Formula: see text] gravity scenario and we determine the connection between the deceleration, jerk, and snap parameters using the flat Friedmann–Robertson–Walker metric and also with the help of the model [Formula: see text], verify Null, Weak, Dominant and Strong energy condition.

In this paper, we concentrate on a new Finsler space that is a combination of two types of Cartan derivatives of third order for Berwald curvature tensor H_jkh^i. We find the condition that Berwald curvature tensor H_jkh^i is special generalized of generalized mixed trirecurrent. Furthermore, we show that the normal projective curvature tensor N_jkh^i is generalized hv- mixed trirecurrent if and only if the tensor ∂ ̇_j (H_hk-H_kh) behaves as mixed trirecurrent. Also, the Ricci tensor N_jk of the normal projective curvature tensor N_jkh^i is non – vanishing if and only if the tensor ((1-n) ∂ ̇_j ∂ ̇_k H+H_jk+H_kj) is mixed trirecurrent.

On the equivalence of distributional and synthetic Ricci curvature lower bounds

Ricci curvature discretizations for head pose estimation from a single image

abstract: We consider how a closed Riemannian manifold $M$ and its metric tensor $g$ can be approximately reconstructed from local distance measurements. Moreover, we consider an inverse problem of determining $(M,g)$ from limited knowledge on the heat kernel. In the part 1 of the paper, we considered the approximate construction of a smooth manifold in the case when one is given the noisy distances $\tilde{d}(x,y)=d(x,y)+\varepsilon_{x,y}$ for all points $x,y\in X$, where $X$ is a $\delta$-dense subset of $M$ and $|\varepsilon_{x,y}|<\delta$. In this part 2 of the paper, we consider a similar problem with partial data, that is, the approximate construction of the manifold $(M,g)$ when we are given $\tilde{d}(x,y)$ for $x\in X$ and $y\in U\cap X$, where $U$ is an open subset of $M$. In addition, we consider the inverse problem of determining the manifold $(M,g)$ with non-negative Ricci curvature from noisy observations of the heat kernel $G(y,z,t)$. We show that a manifold approximating $(M,g)$ can be determined in a stable way, when for some unknown source points $z_j$ in $X\setminus U$, we are given the values of the heat kernel $G(y,z_k,t)$ for $y\in X\cap U$ and $t\in (0,1)$ with a multiplicative noise. We also give a uniqueness result for the inverse problem in the case when the data does not contain noise and consider applications in manifold learning. A novel feature of the inverse problem for the heat kernel is that the set $M\setminus U$ containing the sources and the observation set $U$ are disjoint.

In this paper, we prove some generalizations of the Ambrose (or Myers) theorem for complete Riemannian manifolds. We observe that the problem of finding the Ricci curvature conditions that guarantee the compactness of the manifold is reduced to the problem of finding the proper oscillation conditions of second-order linear differential equations. The proof of the theorems is based on the Riccati comparison theorem and some related oscillation conditions.

UDC 515.1 We study geometric aspects of an almost conformal Ricci soliton and an almost conformal gradient Ricci soliton on K-contact manifolds. Among others, we first obtain the nature of almost conformal Ricci soliton under the conditions: (i) the potential vector field is a contact vector field, and (ii) the potential vector field is pointwise collinear with the Reeb vector field $\xi$. Moreover, we present an example of almost conformal Ricci soliton on a K-contact manifold with potential vector field as a contact vector field. We also find a necessary and sufficient condition for the existence of cyclic Ricci tensor on a K-contact manifold. Further, we give a necessary and sufficient condition for the potential vector field $V$ of a conformal Ricci soliton to be Jacobi along $\xi$ on the K-contact $\eta$-Einstein manifold, and study the nature of almost conformal Ricci soliton on the K-contact $\eta$-Einstein manifold when the potential vector field is a conformal vector field. Finally, we prove that if a complete K-contact metric is an almost conformal gradient Ricci soliton, then the manifold is isometric to a hyperbolic space $H^{2n+1}(\,-1)\,$.

We consider semi-reducible pseudo-Riemannian spaces with algebraic conditions on the Ricci tensor and the Riemann tensor. For almost Einstein and weakly recurrent spaces we find the type of tensor characteristic of semi-reducibility. Semi-reducible almost Einstein spaces and weakly recurrent spaces are divided into types depending on the properties of the vector fields that exist in them by necessity. The study is carried out locally in the tensor form.

Abstract We consider the Steklov problem on differential $$p\text {-}$$ p - forms defined by Karpukhin and present geometric eigenvalue bounds in the setting of warped product manifolds in various scenarios. In particular, we obtain Escobar type lower bounds for warped product manifolds with non-negative Ricci curvature and strictly convex boundary, and certain sharp bounds for hypersurfaces of revolution, among others. We compare and contrast the behaviour with known results in the case of functions (i.e., $$0\text {-}$$ 0 - forms), highlighting the influence of the underlying topology on the spectrum for $$p\text {-}$$ p - forms in general.

Ricci-GraphDTA: A graph neural network integrating discrete Ricci curvature for drug-target affinity prediction.