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We construct a rotationally invariant Ricci flow through surgery starting at any closed rotationally invariant Riemannian manifold. We demonstrate that a sequence of such Ricci flows with surgery converges to a Ricci flow spacetime in the sense of Kleiner and Lott [Acta Math. 219 (2017), pp. 65–134]. Results of Bamler-Kleiner [Acta Math. 228 (2022), pp. 1–215] and Haslhofer [Proc. Amer. Math. Soc. 150 (2022), pp. 5433–5437] then guarantee the uniqueness and stability of these spacetimes given initial data. We simplify aspects of this proof in our setting, and show that for rotationally invariant Ricci flows, the closeness of spacetimes can be measured by equivariant comparison maps. Finally we show that the blowup rate of the curvature near a singular time for these Ricci flows is bounded by the inverse of remaining time squared. [ABSTRACT FROM AUTHOR]

We study solutions of a quadratic matrix equation arising in Riemannian geometry. Let S be a real symmetric n × n -matrix with zeros on the diagonal and let θ be a real number. We construct nonzero solutions (S , θ) of the set of quadratic equations ∑ k S i , k = 0 and ∑ k S i , k S k , j + S i , j 2 = θ S i , j for i < j. Our solutions relate the equations to strongly regular graphs, to group rings, and to multiplicative characters of finite fields. [ABSTRACT FROM AUTHOR]

The metric flow is introduced and extensively studied by Bamler [Compactness theory of the space of super Ricci flows, preprint (2020), arXiv:2008.09298; Structure theory of non-collapsed limits of Ricci flows, preprint (2020), arXiv:2009.03243], especially as an -limit of a sequence of smooth Ricci flows with uniformly bounded Nash entropy, in which case each regular point on the limit is a point of smooth convergence. In this note, we shall consider the -convergence of a sequence of -limit flows, and, like Bamler, show that each regular point on the limit is also a point of smooth convergence. The main result will be applied in another work of the authors [P.-Y. Chan, Z. Ma and Y. Zhang, Dimension reduction for positively curved steady solitons, preprint (2023), arXiv:2310.14020]. [ABSTRACT FROM AUTHOR]

This work answers the key problem on the existence of a Finslerian Ricci flow on short time, and its uniqueness in the isotropic case. The paper also includes two illustrative examples to further support the presented results. [ABSTRACT FROM AUTHOR]

A wide class of two‐component evolution systems is constructed admitting an infinite‐dimensional Lie algebra. Some examples of such systems that are relevant to reaction–diffusion systems with cross‐diffusion are highlighted. It is shown that a nonlinear evolution system related to the Ricci flow on warped product manifold, which has been extensively studied by several authors, follows from the above‐mentioned class as a very particular case. The Lie symmetry properties of this system and its natural generalization are identified and a wide range of exact solutions is constructed using the Lie symmetry obtained. Moreover, a special case is identified when the system in question is reducible to the fast diffusion equation in one space dimension. Finally, another class of two‐component evolution systems with an infinite‐dimensional Lie symmetry that possess essentially different structures is presented. [ABSTRACT FROM AUTHOR]

By using the monotonicity of the log Sobolev functionals, we prove a no breathers theorem for noncompact harmonic Ricci flows under conditions on infimum of log Sobolev functionals and curvatures. As an application, we obtain a no breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature. [ABSTRACT FROM AUTHOR]

In this paper, we study the gradient estimate for the positive solutions of the equation Δu + au(logu)p + bu = f on an almost Ricci soliton (Mn, g, X, λ). In a special case, when X = ∇h for a smooth function h, we derive a gradient estimate for an almost gradient Ricci soliton. [ABSTRACT FROM AUTHOR]

In this work, we study and solve the normalized Ricci flow equation for circle bundles over surfaces. Moreover, we study the asymptotic behavior of the solutions and their connections to some model geometries. [ABSTRACT FROM AUTHOR]

The aim of the present article is to characterize (almost) coKähler manifolds admitting η$$ \eta $$‐Ricci‐Bourguignon solitons. Among others, it is proved that if a three‐dimensional compact almost coKähler manifold admits η$$ \eta $$‐Ricci‐Bourguignon soliton and the potential vector field is pointwise collinear with the Reeb vector field, then the manifold is K‐almost coKähler and η$$ \eta $$‐Einstein. Gradient η$$ \eta $$‐Ricci‐Bourguignon solitons on such manifolds are also considered. Finally, we construct an example of almost coKähler manifold to verify a result. [ABSTRACT FROM AUTHOR]

We consider the more general geometrical flow in the space of metrics for three-manifolds that consists of a combination of two flows, the Cotton flow and the Ricci flow; by playing a fundamental role in the detailed balance principle of the four dimensional Hořava–Lifshitz gravity, this generalized flow reveals another difficulty with this theory that attempts to be a candidate for an UV completion of Einstein general relativity, namely, the supposed emergency of the speed of light, the Newton constant, and the cosmological constant, from a deeply nonrelativistic theory of gravity. Respecting that principle, the generalized flow shows the proliferation of different limits of the theory with an unwanted behavior at both the IR and UV regimes. [ABSTRACT FROM AUTHOR]

Let (M , g) be a noncompact, connected, complete Riemannian three-manifold with nonnegative Ricci curvature satisfying Ric ≥ ε ⁢ tr ⁡ (Ric) ⁢ g for some ε > 0 . In this note, we give a new proof based on inverse mean curvature flow that (M , g) is either flat or has non-Euclidean volume growth. In conjunction with the work of J. Lott [On 3-manifolds with pointwise pinched nonnegative Ricci curvature, Math. Ann.388 (2024), 3, 2787–2806] and of M.-C. Lee and P. Topping [Three-manifolds with non-negatively pinched Ricci curvature, preprint (2022), https://arxiv.org/abs/2204.00504], this gives an alternative proof of a conjecture of R. Hamilton recently proven by A. Deruelle, F. Schulze, and M. Simon [Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds, preprint (2022), https://arxiv.org/abs/2203.15313] using Ricci flow. [ABSTRACT FROM AUTHOR]

By using fixed point argument, we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric $g=\frac {da^2}{h(a^2)}+a^2g_{S^n}$ for some function h where $g_{S^n}$ is the standard metric on the unit sphere $S^n$ in $\mathbb {R}^n$ for any $n\ge 2$. More precisely, for any $\lambda \ge 0$ and $c_0>0$ , we prove that there exist infinitely many solutions ${h\in C^2((0,\infty);\mathbb {R}^+)}$ for the equation $2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-\lambda r-(n-1))$ , $h(r)>0$ , in $(0,\infty)$ satisfying $\underset {\substack {r\to 0}}{\lim }\,r^{\sqrt {n}-1}h(r)=c_0$ and prove the higher-order asymptotic behavior of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behavior near the origin. [ABSTRACT FROM AUTHOR]

In this paper, we consider Ricci flows admitting closed and smooth tangent flows in the sense of Bamler (Structure theory of non-collapsed limits of Ricci flows, 2020. arXiv:2009.03243). The tangent flow in question can be either a tangent flow at infinity for an ancient Ricci flow, or a tangent flow at a singular point for a Ricci flow developing a finite-time singularity. Among other things, we prove: (1) that in these cases the tangent flow must be unique, (2) that if a Ricci flow with finite-time singularity has a closed singularity model, then the singularity is of Type I and the singularity model is the tangent flow at the singular point; this answers a question proposed in Chow et al. (The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects. Mathematical surveys and monographs, vol 163. AMS, Providence, 2010), (3) a dichotomy theorem that characterizes ancient Ricci flows admitting a closed and smooth backward sequential limit. [ABSTRACT FROM AUTHOR]

Abstract Deep neural networks (DNNs) are powerful tools for approximating the distribution of complex data. It is known that data passing through a trained DNN classifier undergoes a series of geometric and topological simplifications. While some progress has been made toward understanding these transformations in neural networks with smooth activation functions, an understanding in the more general setting of non-smooth activation functions, such as the rectified linear unit (ReLU), which tend to perform better, is required. Here we propose that the geometric transformations performed by DNNs during classification tasks have parallels to those expected under Hamilton’s Ricci flow—a tool from differential geometry that evolves a manifold by smoothing its curvature, in order to identify its topology. To illustrate this idea, we present a computational framework to quantify the geometric changes that occur as data passes through successive layers of a DNN, and use this framework to motivate a notion of ‘global Ricci network flow’ that can be used to assess a DNN’s ability to disentangle complex data geometries to solve classification problems. By training more than 1500 DNN classifiers of different widths and depths on synthetic and real-world data, we show that the strength of global Ricci network flow-like behaviour correlates with accuracy for well-trained DNNs, independently of depth, width and data set. Our findings motivate the use of tools from differential and discrete geometry to the problem of explainability in deep learning.

In this paper we have found a number of commutator formulas between the weighted divergence, the weighted Laplacian, the weighted Lichnerowicz Laplacian and the Lie derivtive on closed orientable gradient Ricci shrinkers, then using them we have generalized a Theorem of Cao and Zhu about the necessary condition for linear stability of a gradient Ricci shrinker. Recentlly they have also extended our results. [ABSTRACT FROM AUTHOR]

This paper provides a new definition of the Ricci flow on closed manifolds admitting harmonic spinors. It is shown that Perelman’s Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg–Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin–Thorpe inequality for simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exotic K3 surface must become singular. [ABSTRACT FROM AUTHOR]

Lai (Geom Topol 25:3629–3690, 2021) used singular Ricci flows, introduced by Kleiner and Lott (Acta Math 219(1):65–134, 2017), to construct a nonnegative Ricci curvature Ricci flow g(t) emerging from an arbitrary 3D complete noncompact Riemannian manifold (M 3 , g 0) with nonnegative Ricci curvature. We show g(t) is complete for positive times provided g 0 satisfies a volume ratio lower bound that approaches zero at spatial infinity. Our proof combines a pseudolocality result of Lai (2021) for singular flows, together with a pseudolocality result of Hochard (Short-time existence of the Ricci flow on complete, non-collapsed 3-manifolds with Ricci curvature bounded from below, 2016. ) and Simon and Topping (J Differ Geom 122(3):467–518, 2022) for nonsingular flows. We also show that the construction of complete nonnegative complex sectional curvature flows by Cabezas-Rivas and Wilking (J Eur Math Soc (JEMS) 17(12):3153–3194, 2015) can be adapted here to show g(t) is complete for positive times provided g 0 is a compactly supported perturbation of a nonnegative sectional curvature metric. [ABSTRACT FROM AUTHOR]

In this paper, we will focus our attention on the structure of halmost Ricci solitons on complex hyperbolic quadric. We will prove non-existence a contact real hypersurface in the complex hyperbolic quadric Qm∗,m ≥ 3, admitting the gradient almost Ricci soliton. Moreover, the gradient almost Ricci soliton function f is trivial. [ABSTRACT FROM AUTHOR]

In the present paper, we initiate the study of ∗ - η -Ricci soliton within the framework of Kenmotsu manifolds as a characterization of Einstein metrics. Here we display that a Kenmotsu metric as a ∗ - η -Ricci soliton is Einstein metric if the soliton vector field is contact. Further, we have developed the characterization of the Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies gradient almost ∗ - η -Ricci soliton. Next, we deliberate ∗ - η -Ricci soliton admitting (κ , μ) ′ -almost Kenmotsu manifold and proved that the manifold is Ricci flat and is locally isometric to H n + 1 (- 4) × R n . Finally we present some examples to decorate the existence of ∗ - η -Ricci soliton, gradient almost ∗ - η -Ricci soliton on Kenmotsu manifold. [ABSTRACT FROM AUTHOR]

The main purpose of the present paper is about characterizing properties of the perfect uid space-time admits the gradient Ricci- Bourguignon soliton. This gives some results about stability of the energy momentum tensor and also under some conditions pursues that a perfect uid space-time is Ricci symmetric. As an special case, when a perfect uid space-time equipped with the Ricci-Bourguignon soliton which has Ricci biconformal vector field, we show that the metric of this space is Einstein. [ABSTRACT FROM AUTHOR]

We introduce a classification conjecture for κ-solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new one-parameter family of ℤ 2 2 × O 3 -symmetric bubble-sheet ovals that we construct. We observe that some special cases of the conjecture follow from recent results in the literature. We also introduce a stronger variant of the classification conjecture for ancient asymptotically cylindrical 4d Ricci flows, which does not assume smoothness and nonnegative curvature operator a priori. Assuming this stronger variant holds true, we establish a canonical neighborhood theorem for 4d Ricci flow through cylindrical singularities, which shares some elements in common with Perelman's canonical neighborhood theorem for 3d Ricci flow as well as the mean-convex neighborhood theorem for mean curvature flow through neck-singularities. Finally, we argue that quotient-necks lead to new phenomena, and sketch an example of non-uniqueness for 4d Ricci flow through singularities. [ABSTRACT FROM AUTHOR]

We show that, for any n\geq 2, there exists a homogeneous space of dimension d=8n-4 with metrics of Ric_{\frac {d}{2}-5}>0 if n\neq 3 and Ric_6>0 if n=3 which evolve under the Ricci flow to metrics whose Ricci tensor is not (d-4)-positive. Consequently, Ricci flow does not preserve a range of curvature conditions that interpolate between positive sectional and positive scalar curvature. This extends a theorem of Böhm and Wilking [Geom. Funct. Anal. 17 (2007), pp. 665–681] in the case of n=2. [ABSTRACT FROM AUTHOR]

In previous work (Topping and Yin, https://arxiv.org/abs/2107.14686), we established the existence of a Ricci flow starting with a Riemann surface coupled with a non‐atomic Radon measure as a conformal factor. In this paper, we prove uniqueness, settling Conjecture 1.3 of Topping and Yin (https://arxiv.org/abs/2107.14686). Combining these two works yields a canonical smoothing of such rough surfaces that also regularises their geometry at infinity. [ABSTRACT FROM AUTHOR]

This article aims to understand the behavior of the curvature operator of the second kind under the Ricci flow in dimension three. First, we express the eigenvalues of the curvature operator of the second kind explicitly in terms of that of the curvature operator (of the first kind). Second, we prove that α -positive/ α -nonnegative curvature operator of the second kind is preserved by the Ricci flow in dimension three for all α ∈ [ 1 , 5 ] . [ABSTRACT FROM AUTHOR]

We establish a structure result for the isometry group of non-compact, homogeneous manifolds admitting an immortal homogeneous Ricci flow solution. [ABSTRACT FROM AUTHOR]

In this note, we prove a well-known conjecture on the Ricci flow under a curvature condition, which is a pinching between the Ricci and Weyl tensors divided by suitably translated scalar curvature, motivated by Cao’s result (Commun Anal Geom 19(5):975–990, 2011). [ABSTRACT FROM AUTHOR]

We prove sharp lower bounds for eigenvalues of the drift Laplacian for a modified Ricci flow. The modified Ricci flow is a system of coupled equations for a metric and weighted volume that plays an important role in Ricci flow. We will also show that there is a splitting theorem in the case of equality.

Motivated by an extension to Gallavotti–Cohen Chaotic Hypothesis, we study local and global existence of a gradient flow of the Sinai–Ruelle–Bowen entropy functional in the space of transitive Anosov maps. For the space of expanding maps on the unit circle, we equip it with a Hilbert manifold structure using a Sobolev norm in the tangent space of the manifold. Under the additional measure-preserving assumption and a slightly modified metric, we show that the gradient flow exists globally and every trajectory of the flow converges to a unique limiting map where the SRB entropy attains the maximal value. In a simple case, we obtain an explicit formula for the flow's ordinary differential equation representation. This gradient flow has close connection to a nonlinear partial differential equation, a gradient-dependent diffusion equation. [ABSTRACT FROM AUTHOR]

In this paper, we determine the variation formula for the first eigenvalue of (p , q) -biharmonic system on a closed Riemannian manifold. Several monotonic quantities are also derived. [ABSTRACT FROM AUTHOR]

We introduce a novel curvature flow, the Heterotic-Ricci flow, as the two-loop renormalization group flow of the Heterotic string common sector and study its three-dimensional compact solitons. The Heterotic-Ricci flow is a coupled curvature evolution flow, depending on a non-negative real parameter κ , for a complete Riemannian metric and a three-form H on a manifold M. Its most salient feature is that it involves several terms quadratic in the curvature tensor of a metric connection with skew-symmetric torsion H. When κ = 0 the Heterotic-Ricci flow reduces to the generalized Ricci flow and hence it can be understood as a modification of the latter via the second-order correction prescribed by Heterotic string theory, whereas when H = 0 and κ > 0 the Heterotic-Ricci flow reduces to a constrained version of the RG-2 flow and hence it can be understood as a generalization of the latter via the introduction of the three-form H. Solutions of Heterotic supergravity with trivial gauge bundle, which we call Heterotic solitons, define a particular class of three-dimensional solitons for the Heterotic-Ricci flow and constitute our main object of study. We prove a number of structural results for three-dimensional Heterotic solitons, obtaining, in particular, the complete classification of compact three-dimensional strong Heterotic solitons as hyperbolic three-manifolds or quotients of the Heisenberg group equipped with a left-invariant metric. Furthermore, we prove that all Einstein three-dimensional Heterotic solitons have constant dilaton and leave as open the construction of a Heterotic soliton with non-constant dilaton. In this direction, we prove that Einstein Heterotic solitons with constant dilaton are rigid and therefore cannot be deformed into a solution with non-constant dilaton. This is, to the best of our knowledge, the first rigidity result for compact supergravity solutions in the literature. [ABSTRACT FROM AUTHOR]

Ancient solutions of the Ricci flow arise naturally as models for singularity formation. There has been significant progress towards the classification of such solutions under natural geometric assumptions. Nonnegatively curved solutions in dimensions 2 and 3, and uniformly PIC solutions in higher dimensions are now well understood. We consider ancient solutions of arbitrary dimension which are complete and have Type I curvature growth. We show that a simply connected, noncompact, κ -noncollapsed Type I ancient solution with nonnegative sectional curvature necessarily splits a Euclidean factor. It follows that a κ -noncollapsed Type I ancient solution which is weakly PIC2 is a locally symmetric space. [ABSTRACT FROM AUTHOR]

We study singularity formation for the heat flow of harmonic maps from R d . For each d ≥ 4 , we construct a compact, d-dimensional, rotationally symmetric target manifold that allows for the existence of a corotational self-similar shrinking solution (shortly shrinker) that represents a stable blowup mechanism for the corresponding Cauchy problem. [ABSTRACT FROM AUTHOR]

In this paper, we have established the existence of conformal Ricci soliton as self similar solution of conformal Ricci flow and characterized the potential function f of almost conformal gradient Ricci soliton on a Sasakian manifold in terms of the non-dynamical scalar field p. We have found the necessary condition for the potential function f to be constant. Furthermore, a relation between λ and the potential function f has been established. Finally, we prove a sufficient condition for an almost conformal Ricci soliton to be an almost conformal gradient Ricci soliton and characterize it in terms of shrinking, steady or expanding soliton. [ABSTRACT FROM AUTHOR]

Community structure is a fundamental topological characteristic of optimally organized brain networks. Currently, there is no clear standard or systematic approach for selecting the most appropriate community detection method. Furthermore, the impact of method choice on the accuracy and robustness of estimated communities (and network modularity), as well as method‐dependent relationships between network communities and cognitive and other individual measures, are not well understood. This study analyzed large datasets of real brain networks (estimated from resting‐state fMRI from n$$ n $$ = 5251 pre/early adolescents in the adolescent brain cognitive development [ABCD] study), and n$$ n $$ = 5338 synthetic networks with heterogeneous, data‐inspired topologies, with the goal to investigate and compare three classes of community detection methods: (i) modularity maximization‐based (Newman and Louvain), (ii) probabilistic (Bayesian inference within the framework of stochastic block modeling (SBM)), and (iii) geometric (based on graph Ricci flow). Extensive comparisons between methods and their individual accuracy (relative to the ground truth in synthetic networks), and reliability (when applied to multiple fMRI runs from the same brains) suggest that the underlying brain network topology plays a critical role in the accuracy, reliability and agreement of community detection methods. Consistent method (dis)similarities, and their correlations with topological properties, were estimated across fMRI runs. Based on synthetic graphs, most methods performed similarly and had comparable high accuracy only in some topological regimes, specifically those corresponding to developed connectomes with at least quasi‐optimal community organization. In contrast, in densely and/or weakly connected networks with difficult to detect communities, the methods yielded highly dissimilar results, with Bayesian inference within SBM having significantly higher accuracy compared to all others. Associations between method‐specific modularity and demographic, anthropometric, physiological and cognitive parameters showed mostly method invariance but some method dependence as well. Although method sensitivity to different levels of community structure may in part explain method‐dependent associations between modularity estimates and parameters of interest, method dependence also highlights potential issues of reliability and reproducibility. These findings suggest that a probabilistic approach, such as Bayesian inference in the framework of SBM, may provide consistently reliable estimates of community structure across network topologies. In addition, to maximize robustness of biological inferences, identified network communities and their cognitive, behavioral and other correlates should be confirmed with multiple reliable detection methods. [ABSTRACT FROM AUTHOR]

We investigate geometric aspects of complete spacelike mean curvature flow solitons of codimension one in a generalized Robertson–Walker (GRW) spacetime $-I\times _{f}M^n$ , with base $I\subset \mathbb R$ , Riemannian fiber $M^n$ and warping function $f\in C^\infty (I)$. For this, we apply suitable maximum principles to guarantee that such a mean curvature flow soliton is a slice of the ambient space and to obtain nonexistence results concerning these solitons. In particular, we deal with entire graphs constructed over the Riemannian fiber $M^n$ , which are spacelike mean curvature flow solitons, and we also explore the geometry of a conformal vector field to establish topological and further rigidity results for compact (without boundary) mean curvature flow solitons in a GRW spacetime. Moreover, we study the stability of spacelike mean curvature flow solitons with respect to an appropriate stability operator. Standard examples of spacelike mean curvature flow solitons in GRW spacetimes are exhibited, and applications related to these examples are given. [ABSTRACT FROM AUTHOR]

In this paper, the class of homogeneous Siklos space-times is considered from algebraic point of view and the generalized Ricci solitons are completely classified. [ABSTRACT FROM AUTHOR]

In this study, we demonstrate that when three distinct moving space-like curves with the time-like binormal vectors satisfy the generalizations of the Heisenberg spin chain model, these models are equivalent to the third-order system of the AKNS hierarchy and the complex KdV-type equations. [ABSTRACT FROM AUTHOR]

Complex biological processes, such as cellular differentiation, require intricate rewiring of intra-cellular signalling networks. Previous characterisations revealed a raised network entropy underlies less differentiated and malignant cell states. A connection between entropy and Ricci curvature led to applications of discrete curvatures to biological networks. However, predicting dynamic biological network rewiring remains an open problem. Here we apply Ricci curvature and Ricci flow to biological network rewiring. By investigating the relationship between network entropy and Forman-Ricci curvature, theoretically and empirically on single-cell RNA-sequencing data, we demonstrate that the two measures do not always positively correlate, as previously suggested, and provide complementary rather than interchangeable information. We next employ Ricci flow to derive network rewiring trajectories from stem cells to differentiated cells, accurately predicting true intermediate time points in gene expression time courses. In summary, we present a differential geometry toolkit for understanding dynamic network rewiring during cellular differentiation and cancer. When stem cells develop into tissues intracellular signalling is rewired, errors in this process lead to cancer. Here, authors applied tools from differential geometry made by Albert Einstein's General Relativity to understand and predict biological network rewiring in health and disease. [ABSTRACT FROM AUTHOR]

Assuming Perelman's estimates, we give a new proof of uniform L^\infty estimate along normalized Kähler-Ricci flow on Fano manifolds with Kähler-Einstein metrics, using Chen-Cheng's auxiliary Monge-Ampère equation and the Alexandrov-Bakelman-Pucci maximum principle. This proof does not use pluripotential theory. [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]

In Glickenstein (J Differ Geom 87: 201–237, 2011), Glickenstein introduced the discrete conformal structures on polyhedral surfaces in an axiomatic approach from Riemannian geometry perspective. It includes Thurston's circle packings, Bowers–Stephenson's inversive distance circle packings and Luo's vertex scalings as special cases. In this paper, we study the deformation of Glickenstein's discrete conformal structures by combinatorial curvature flows. The combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces (Zhang et al. in Graph Models 76: 321–339, 2014) is a generalization of Chow–Luo's combinatorial Ricci flow for Thurston's circle packings (Chow and Luo in J Differ Geom 63: 97–129, 2003) and Luo's combinatorial Yamabe flow for vertex scalings (Luo in Commun Contemp Math 6: 765–780, 2004). We prove that the solution of the combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces can be uniquely extended. Furthermore, under some necessary conditions, we prove that the solution of the extended combinatorial Ricci flow on a triangulated surface exists for all time and converges exponentially fast for any initial value. We further introduce the combinatorial Calabi flow for Glickenstein's discrete conformal structures on triangulated surfaces and study the basic properties of the flow. These combinatorial curvature flows provide effective algorithms for finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures. [ABSTRACT FROM AUTHOR]

We prove that the bi-invariant Einstein metric on |$SU_{2n+1}$| is isolated in the moduli space of Einstein metrics, even though it admits infinitesimal deformations. This gives a non-Kähler, non-product example of this phenomenon adding to the famous example of |$\mathbb{C}\mathbb{P}^{2n}\times \mathbb{C}\mathbb{P}^{1}$| found by Koiso. We apply our methods to derive similar solitonic rigidity results for the Kähler–Einstein metrics on "odd" Grassmannians. We also make explicit a connection between non-integrable deformations and the dynamical instability of metrics under Ricci flow. [ABSTRACT FROM AUTHOR]

In this note, we prove that positive scalar curvature can pass to three dimensional Ricci limit spaces of non-negative Ricci curvature when it splits off a line. As a corollary, we obtain an optimal Bonnet-Myers type upper bound. Moreover, we obtain a similar statement in all dimensions for Alexandrov spaces of non-negative curvature. [ABSTRACT FROM AUTHOR]

The process of 3D reconstruction involves stacking 2D scans of medical images to create a mesh of vertices and edges with volumetric details about the organ of interest. However, sometimes, the resulting surface may contain noise and rough texture. To address this, mesh smoothing algorithms are used, with many employing a central-vertex-first approach where the vertex under observation is updated based on features of its neighbors. However, the limitation of central-vertex-first philosophy is that it does not consider a patch as a single entity during the process of smoothing and rather treats every single vertex as an entity which in turn leads to starvation. This paper presents a novel all-neighbor-first approach to smoothen the reconstructed human organs using a target-based approach, based on Ricci Flow and circle packing metric. The proposed model allows for the alteration of noisy surface contours to match the final result, and can be modified to generalize for any surface whose target is not known. The paper proposes two different schemes for updating the vertices based on RF metrics. Importantly, the proposed model is teacher-based and requires no training examples, saving computational time compared to supervised Deep Learning methods. Overall, this approach offers a unique and promising solution to smoothing 3D reconstructions with noisy surfaces, with potential applications in medical imaging and beyond. [ABSTRACT FROM AUTHOR]

We extend the anholonomic frame and connection deformation method, AFCDM, for constructing exact and parametric solutions in general relativity, GR, to geometric flow models and modified gravity theories, MGTs, with nontrivial torsion and nonmetricity fields. Following abstract geometric or variational methods, we can derive corresponding systems of nonmetric gravitational and matter field equations which consist of very sophisticate systems of coupled nonlinear PDEs. Using nonholonomic frames with dyadic spacetime splitting and applying the AFCDM, we prove that such systems of PDEs can be decoupled and integrated in general forms for generic off-diagonal metric structures and generalized affine connections. We generate new classes of quasi-stationary solutions (which do not depend on time like coordinates) and study the physical properties of some physically important examples. Such exact or parametric solutions are determined by nonmetric solitonic distributions and/or ellipsoidal deformations of wormhole hole configurations. It is not possible to describe the thermodynamic properties of such solutions in the framework of the Bekenstein–Hawking paradigm because such metrics do not involve, in general, certain horizons, duality, or holographic configurations. Nevertheless, we can always elaborate on associated Grigori Perelman thermodynamic models elaborated for nonmetric geometric flows. In explicit form, applying the AFCDM, we construct and study the physical implications of new classes of traversable wormhole solutions describing solitonic deformation and dissipation of non-Riemannian geometric objects. Such models with nontrivial gravitational off-diagonal vacuum are important for elaborating models of dark energy and dark matter involving wormhole configurations and solitonic-type structure formation. [ABSTRACT FROM AUTHOR]

In this paper, we study complete Type I ancient Ricci flows with positive sectional curvature. Our main results are as follows: in the complete and noncompact case, all such ancient solutions must be noncollapsed on all scales; in the closed case, if the dimension is even, then all such ancient solutions must be noncollapsed on all scales. This furthermore gives a complete classification for three‐dimensional noncompact Type I ancient solutions without assuming the noncollapsing condition. [ABSTRACT FROM AUTHOR]

We show that there exists a quantity, depending only on C 0 data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the C 0 sense and has nonnegative scalar curvature in the sense of Ricci flow. Moreover, the C 0 mass at infinity is independent of choice of C 0 -asymptotically flat coordinate chart, and the C 0 local mass has controlled distortion under Ricci–DeTurck flow when coupled with a suitably evolving test function. [ABSTRACT FROM AUTHOR]

We prove that the Ricci flow for complete metrics with bounded geometry depends continuously on initial conditions for finite time with no loss of regularity. This relies on recent work of Bahuaud, Guenther, Isenberg and Mazzeo where sectoriality for the generator of the Ricci-DeTurck flow is proved. We use this to prove that for initial metrics sufficiently close in Hölder norm to a rotationally symmetric asymptotically hyperbolic metric and satisfying a simple curvature condition, but a priori distant from the hyperbolic metric, Ricci flow converges to the hyperbolic metric. [ABSTRACT FROM AUTHOR]