Search Results (37)

In this paper, we study the Ricci-harmonic flow under the assumption that the scalar curvature is bounded. First, we establish a time-derivative bound for solutions to the heat equation along the flow. Based on this estimate, we derive a short-time distance-distortion estimate and prove the existence of suitable cutoff functions. Using these results, we obtain Gaussian-type upper and lower bounds for the heat kernel along the Ricci-harmonic flow. Our results generalize the previous work of Bamler–Zhang on Ricci flow to the Ricci-harmonic flow setting, and can be used to study the regularity theory of Ricci-harmonic flow. Full article

The present paper is aimed at studying the convergence of the Yamabe flow in the case of noncompact solitons. The more specified example of locally conformally flat noncompact solitons is addressed with the aim to newly analyse the qualities of the Ricci scalar. The particular case of noncompact pseudo-Riemannian solitons is studied; moreover, in the instances of Schwarzschild and Generalized-Schwarzschild geometries, rescalings of spherically symmetric weights are performed. For this purpose, new results are achieved as far as the considered structures are concerned. The Myers Theorem is upgraded as the new Myers paradigm of spacetime-dimensional manifolds, where the Einstein Field Equations can now be taken into account. In particular, the Myers Theorems are studied here as far as their new implementation in General Relativity Theory is concerned. As a first important result, the Myers mean curvature is found to coincide with the Ricci scalar in General Relativity Theory, where the 4-position of the observer, from which the 4-velocity 4-vector is calculated from, is taken as that of the observer solidal with the reference frame of the photon. The following results are also of relevance. In more detail, the umbilicity conditions are applied. At a further step, the role of the umbilicity conditions in GR after the Myers Theorems are studied for weighted manifolds and specific new implications of weighted manifolds are developed. The description of the weighted Schwarzschild manifolds and that of the weighted Generalized-Schwarzschild manifolds are newly studied as follows: as a new finding, the Birkhoff Theorem is newly reconciled with the rotational ansatz of the metrised solitons, and the comparison with the previous results about the Brendle non-metrised solitons is accomplished with the outcome stressing the new roles of the new rescalings of the metric tensor with respect to the previous known results of the scaling of the metric tensor of the non-metrised solitons. In the present framework, these procedures allow one to prove the reconciliation of the EFEs with the Yamabe flow. The flow on the tipping lightcones is newly written. The umbilicity condition is studied in General Relativity after the upgrade of the Myers Theorems as far as the sectional curvatures are concerned; as a result, the Calabi–Bernstein description is implemented in General Relativity, as well as the Chen–Yau requirements, and the cases of weighted manifolds are taken into account. More specifically, the equal-time 2-dimensional space surfaces are studied analytically, onto which the weighted General-Relativistic solitons which satisfy the Einstein field equations after the Yamabe flow are projected due to the rotational ansatz. As an accessory introductory result, the class of Wu non-metrised solitons are proven to be discarded in several aspects of the Wu description as the conditions provided after the work of Wu are not compatible with metrisation. Full article

We develop an informational extension of spacetime thermodynamics in which local entropy production is coupled to spacetime curvature within an effective covariant framework. Spacetime is modeled as a continuum limit of finite-capacity information registers, giving rise to a coarse-grained entropy field whose gradients define an informational flux. Within a nonminimally coupled scalar–tensor formulation, the resulting field equations imply that the local divergence of this flux is sourced by the Ricci scalar, establishing a direct relation between curvature and entropy production. The corresponding integral form links cumulative entropy generation to the integrated spacetime curvature over a causal region. In stationary limits, the framework reproduces the Bekenstein–Hawking entropy of horizons, while in homogeneous expanding cosmologies it yields monotonic entropy growth consistent with the observed arrow of time. The construction remains compatible with unitarity at the microscopic level and with holographic entropy bounds in the stationary limit. Numerical solutions in flat FLRW backgrounds are used as consistency checks of the coupled evolution equations and confirm the expected curvature–entropy behavior across cosmological epochs. Overall, the results provide a thermodynamically consistent interpretation of curvature as a geometric source of irreversible information flow, without modifying the underlying gravitational field equations. Full article

In freeform optical metrology, wavefront fitting over non-circular apertures is hindered by the loss of Zernike polynomial orthogonality and severe sampling grid distortion inherent in standard conformal mappings. To address the resulting numerical instability and fitting bias, we propose a unified framework curve-shortening flow (CSF)-guided progressive quasi-conformal mapping (CSF-QCM), which integrates geometric boundary evolution with topology-aware parameterization. CSF-QCM first smooths complex boundaries via curve-shortening flow, then solves a sparse Laplacian system for harmonic interior coordinates, thereby establishing a stable diffeomorphism between physical and canonical domains. For doubly connected apertures, it preserves topology by computing the conformal modulus via Dirichlet energy minimization and simultaneously mapping both boundaries. Benchmarked against state-of-the-art methods (e.g., Fornberg, Schwarz–Christoffel, and Ricci flow) on representative irregular apertures, CSF-QCM suppresses area distortion and restores discrete orthogonality of the Zernike basis, reducing the Gram matrix condition number from >900 to <8. This enables high-precision reconstruction with RMS residuals as low as $3\times {10}^{-3}\lambda$ and up to 92% lower fitting errors than baselines. The framework provides a unified, computationally efficient, and numerically stable solution for wavefront reconstruction in complex off-axis and freeform optical systems. Full article

The historical and conceptual foundations of General Relativity are revisited, putting the main focus on the physical meaning of the invariant $d{s}^2$, the Equivalence Principle, and the precise interpretation of spacetime geometry. It is argued that Albert Einstein initially sought a dynamical formulation in which $d{s}^2$ encoded the gravitational effects, without invoking curvature as a physical entity. The now more familiar geometrical interpretation—identifying gravitation with spacetime curvature—gradually emerged through his collaboration with Marcel Grossmann and the adoption of the Ricci tensor in 1915. Anyhow, in his 1920 Leiden lecture, Einstein explicitly reinterpreted spacetime geometry as the state of a physical medium—an “ether” endowed with metrical properties but devoid of mechanical substance—thereby actually rejecting geometry as an independent ontological reality. Building upon this mature view, gravitation is reconstructed from the Weak Equivalence Principle, understood as the exact compensation between inertial and gravitational forces acting on a body under a uniform gravitational field. From this fundamental principle, together with an extension of Fermat’s Principle to massive objects, the invariant $d{s}^2$ is obtained, first in the static case, where the gravitational potential modifies the flow of proper time. Then, by applying the Lorentz transformation to this static invariant, its general form is derived for the case of matter in motion. The resulting invariant reproduces the relativistic form of Newton’s second law in proper time and coincides with the weak-field limit of General Relativity in the harmonic gauge. This approach restores the operational meaning of Einstein’s theory: spacetime geometry represents dynamical relations between physical measurements, rather than the substance of spacetime itself. By deriving the gravitational modification of the invariant directly from the Weak Equivalence Principle, Fermat Principle and Lorentz invariance, this formulation clarifies the physical origin of the metric structure and resolves long-standing conceptual issues—such as the recurrent hole argument—while recovering all the empirical successes of General Relativity within a coherent and sound Machian framework. Full article

In this paper, we study the topology of steady gradient Ricci solitons with nonnegative sectional curvature. We apply a characterization theorem for the fundamental group of a positively curved steady gradient Ricci soliton that admits a critical point. As an application of the characterization theorem, we give a classification of the topology of complete four-dimensional steady gradient Ricci solitons with nonnegative sectional curvature. Full article

Let $(M,g)$ be a connected, compact Riemannian manifold of dimensionan n. We demonstrate that, after a suitable normalization, a shrinking gradient Ricci soliton $(M,g,f,\lambda )$ is trivial exactly when the mean value of f is less than or equal to $\frac{n}{2}$. Moreover, we prove that a normalized non-steady gradient Ricci soliton $(M,g,f,\lambda )$ is trivial if and only if its scalar curvature S satisfies the relation $S=\lambda \left(f+\frac{n}{2}\right).$ In addition, we establish that if $(M,g,f,\lambda )$ admits an isometric immersion as a hypersurface in the Euclidean space, then the soliton must necessarily be of a shrinking type. In such a case, the constant $\lambda$ and the mean curvature of M satisfy a certain inequality, with equality occurring precisely when M is isometric to a round sphere. Full article

In this paper, the uniqueness of steady Schwarzschild gradient Ricci solitons is studied. The role of the weight functions is analyzed. The generalized steady Schwarzschild gradient Ricci solitons are investigated; the implications of the rotational ansatz of Bryant are developed; and the new Generalized Schwarzschildsteady gradient solitons are defined. The aspects of the weight functions of the latter type of solitons are researched as well. The new most-accurate curvature bound of the steady Ricci gradient solitons is provided. The uniqueness of the Schwarzschild solitons is discussed. The Ricci flow is reconciled with the Einstein Field Equations such that the weight functions are utilized to spell out the determinant of the metric tensor, the procedure for which is commented on following the use of the appropriate geometrical objects. The mean curvature is discussed. The configurations of the observer are issued from the geodesics spheres of the solitonic structures. Full article

Looking at current proposals of so-called ‘warp drive spacetimes’, they appear to employ General Relativity only at an elementary level. A number of strong restrictions are imposed such as flow-orthogonality of the spacetime foliation, vanishing spatial Ricci tensor, and dimensionally reduced and coordinate-dependent velocity fields, to mention the main restrictions. We here provide a brief summary of our proposal of a general and covariant description of spatial motions within General Relativity, then discuss the restrictions that are employed in the majority of the current literature. That current warp drive models are discussed to be unphysical may not be surprising; they lack important ingredients such as covariantly non-vanishing spatial velocity, acceleration, vorticity together with curved space, and a warp mechanism. Full article

The Fermi condensation flows in the sine-Gordon–Thirring model with two impurities coupling are investigated in this paper; these matter flows can be induced by the Ricci flow perturbation in the two-dimensional string $\sigma$ model. The Ricci flow perturbation equations are derived according to the Gauss–Codazzi equations, and the two-loop asymptotic perturbation solution of the cigar soliton is reduced by using a small parameter expansion method. Moreover, the thermodynamic quantities on the cigar soliton background are obtained by using the variational functional integrals method. Subsequently, the Fermi condensation flows varying with the momentum scale $\lambda$ are analyzed and discussed. We find that the energy density, the correlation function, and the condensation fluctuations will decrease, but the entropy will increase monotonically. The Fermi condensed matter can maintain thermodynamic stability under the Ricci flow perturbation. Full article

This paper undertakes a detailed study of $\eta$-Ricci–Bourguignon solitons on $ϵ$-Kenmotsu manifolds, with particular focus on three special types of Ricci tensors: Codazzi-type, cyclic parallel and cyclic $\eta$-recurrent tensors that support such solitonic structures. We derive key curvature conditions satisfying Ricci semi-symmetric $(R·E=0)$, conharmonically Ricci semi-symmetric $(C(\xi ,{\beta }_X)·E=0)$, $\xi$-projectively flat $(P({\beta }_X,{\beta }_Y)\xi =0)$, projectively Ricci semi-symmetric $(L·P=0)$ and $W_5$-Ricci semi-symmetric $(W(\xi ,{\beta }_Y)·E=0)$, respectively, with the admittance of $\eta$-Ricci–Bourguignon solitons. This work further explores the role of torse-forming vector fields and provides a thorough characterization of $\varphi$-Ricci symmetric indefinite Kenmotsu manifolds admitting $\eta$-Ricci–Bourguignon solitons. Through in-depth analysis, we establish significant geometric constraints that govern the behavior of these manifolds. Finally, we construct explicit examples of indefinite Kenmotsu manifolds that satisfy the $\eta$-Ricci–Bourguignon solitons equation, thereby confirming their existence and highlighting their unique geometric properties. Moreover, these solitonic structures extend soliton theory to indefinite and physically meaningful settings, enhance the classification and structure of complex geometric manifolds by revealing how contact structures behave under advanced geometric flows and link the pure mathematical geometry to applied fields like general relativity. Furthermore, $\eta$-Ricci–Bourguignon solitons provide a unified framework that deepens our understanding of geometric evolution and structure-preserving transformations. Full article

The manifolds studied are almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds. They are equipped with a pair of pseudo-Riemannian metrics that are mutually associated to each other using an almost contact structure. Furthermore, the structural endomorphism acts as an anti-isometry for these metrics, called B-metrics, if its action is restricted to the contact distribution of the manifold. In this paper, some curvature properties of a special class of these manifolds, called Sasaki-like, are studied. Such a manifold is defined by the condition that its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). Each of the two B-metrics on the considered manifold is specialized here as an $\eta$-Ricci–Bourguignon almost soliton, where $\eta$ is the contact form, i.e., has an additional curvature property such that the metric is a self-similar solution of a special intrinsic geometric flow. Almost solitons are generalizations of solitons because their defining condition uses functions rather than constants as coefficients. The introduced (almost) solitons are a generalization of some well-known (almost) solitons (such as those of Ricci, Schouten, and Einstein). The soliton potential is chosen to be collinear with the Reeb vector field and is therefore called vertical. The special case of the soliton potential being solenoidal (i.e., divergence-free) with respect to each of the B-metrics is also considered. The resulting manifolds equipped with the pair of associated $\eta$-Ricci–Bourguignon almost solitons are characterized geometrically. An example of arbitrary dimension is constructed and the properties obtained in the theoretical part are confirmed. Full article

Recent interest among geometers in f-structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric structures introduced by the author and R. Wolak as a generalization of Hermitian and Kähler structures, as well as f-structures, allow for a fresh perspective on the classical theory. In this paper, we study a new f-structure of this kind, called the weak $\beta$-Kenmotsu f-structure, as a generalization of K. Kenmotsu’s concept. We prove that a weak $\beta$-Kenmotsu f-manifold is a locally twisted product of the Euclidean space and a weak Kähler manifold. Our main results show that such manifolds with $\beta =const$ and equipped with an $\eta$-Ricci soliton structure whose potential vector field satisfies certain conditions are $\eta$-Einstein manifolds of constant scalar curvature. Full article

In this paper, an underlying perturbed Ricci flow construction is made within the metric operator space, originating from the Heisenberg dynamical equations, to formulate a method which appears to provide a new geometric approach for the geometric formulation of the quantum mechanical dynamics. A quantum mechanical notion of stability and local instability is introduced within the quantum mechanical theory, based on the quantum mechanical dynamical equations governing the evolution of the tensor metric operator. The stability analysis is conducted in the topology of little $H\ddot{o}lder$ spaces of metrics which the tensor metric operator acts on. Finally, a theorem is introduced in an attempt to characterize the stability properties of the quantum mechanical system such that it brings the quantum mechanical dynamics into the analysis. Full article

Let $\lambda \left(t\right)$ be the first eigenvalue of the operator $-∆+a{R}^b$ on locally three-dimensional homogeneous manifolds along the backward Ricci flow, where $a,b$ are real constants and R is the scalar curvature. In this paper, we study the properties of $\lambda \left(t\right)$ on Bianchi classes. We begin by deriving an evolution equation for the quantity $\lambda \left(t\right)$ on three-dimensional homogeneous manifolds in the context of the backward Ricci flow. Utilizing this equation, we subsequently establish a monotonic quantity that is contingent upon $\lambda \left(t\right)$. Additionally, we present both upper and lower bounds for $\lambda \left(t\right)$ within the framework of Bianchi classes. Full article