SOME PARABOLIC AND ELLIPTIC PROBLEMS IN COMPLEX RIEMANNIAN GEOMETRY

We show that Perelman’s \({\mathcal{W}}\) functional on Kahler manifolds has a natural counterpart on Sasaki manifolds. We prove, using this functional, that Perelman’s results on Kahler–Ricci flow (the first Chern class is positive) can be generalized to Sasaki–Ricci flow, including the uniform bound on the diameter and the scalar curvature along the flow. We also show that positivity of transverse bisectional curvature is preserved along Sasaki–Ricci flow, using Bando and Mok’s methods and results in Kahler–Ricci flow. In particular, we show that the Sasaki–Ricci flow converges to a Sasaki–Ricci soliton when the initial metric has nonnegative transverse bisectional curvature.

We prove that on Fano manifolds, the Kahler–Ricci flow produces a “most destabilising” degeneration, with respect to a new stability notion related to the $H$-functional. This answers questions of Chen–Sun–Wang and He. We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the supremum of Perelman’s $\mu$‑functional on Fano manifolds, resolving a conjecture of Tian–Zhang-Zhang-Zhu as a special case. Secondly, we use this to prove that if a Fano manifold admits a Kahler–Ricci soliton, then the Kahler–Ricci flow converges to it modulo the action of automorphisms, with any initial metric. This extends work of Tian–Zhu and Tian–Zhang–Zhang–Zhu, where either the manifold was assumed to admit a Kahler–Einstein metric, or the initial metric of the flow was assumed to be invariant under a maximal compact group of automorphism.

We use the momentum construction of Calabi to study the conical Kahler–Ricci flow on Hirzebruch surfaces with cone angle along the exceptional curve, and show that either the flow Gromov–Hausdorff converges to the Riemann sphere or a single point in finite time, or the flow contracts only the cone divisor to a single point and Gromov–Hausdorff converges to a two dimensional projective orbifold. The limiting behaviour depends only on the cone angle, numerical properties of the initial Kahler class, and the degree of the Hirzebruch surface. This gives the first example of the conical Kahler–Ricci flow contracting the cone divisor to a single point, and shows that the conical flow may contract curves of self-intersection less than $$(-\,1)$$ , as opposed to the smooth Kahler–Ricci flow. At the end, we introduce a conjectural picture of the geometry of finite time non-collapsing singularities of the flow on Kahler surfaces in general.

The Hopf surfaces provide a family of minimal non-Kähler surfaces of class VII on which little is known about the Chern–Ricci flow. We use a construction of Gauduchon–Ornea for locally conformally Kähler metrics on primary Hopf surfaces of class 1 to study solutions of the Chern–Ricci flow. These solutions reach a volume collapsing singularity in finite time, and we show that the metric tensor satisfies a uniform upper bound, supporting the conjecture that the Gromov-Hausdorff limit is isometric to a round\(S^1\). Uniform \(C^{1+\beta }\) estimates are also established for the potential. Previous results had only been known for the simplest examples of Hopf surfaces.

Consider the Kähler–Ricci flow with finite time singularities over any closed Kähler manifold. We prove the existence of the flow limit in the sense of current toward the time of singularity. This answers affirmatively a problem raised by Tian in [New results and problems on Kähler–Ricci flow, Astérisque 322 (2008) 71–92] on the uniqueness of the weak limit from sequential convergence construction. The notion of minimal singularity introduced by Demailly in the study of positive current comes up naturally. We also provide some discussion on the infinite time singularity case for comparison. The consideration can be applied to more flexible evolution equation of Kähler–Ricci flow type for any cohomology class. The study is related to general conjectures on the singularities of Kähler–Ricci flows.

We study the construction and classification of weakly Bochnerflat (WBF) metrics (i.e., Kähler metrics with coclosed Bochner tensor) on compact complex manifolds.A Kähler metric is WBF if and only if its 'normalized' Ricci form is a hamiltonian 2-form: such 2-forms were introduced and studied in previous papers in the series.It follows that WBF Kähler metrics are extremal.We construct many new examples of WBF metrics on projective bundles and obtain a classification of compact WBF Kähler 6-manifolds, extending work by the first three authors on weakly selfdual Kähler 4-manifolds.The constructions are independent of previous papers in the series, but the classification relies on the classification of compact Kähler manifolds with a hamiltonian 2-form [3].

Abstract. R.Hamilton defined Ricci flow as evolution of Rie-mannian metrics satisfying a weak parabolic partial differential equation. We have considered and studied Ricci flow as an inte-gral curve of a certain vector field in the manifold of Riemannianmetrics. In this paper we study properties of this integral curveand find some results on the stability of the solution.

In this note, we will prove that given a sequence of Ricci flows ( M k , g k ⁢ ( t ) ) ${(M_{k},g_{k}(t))}$ of dimension n, t ∈ [ - 1 , 0 ] ${t\in[-1,0]}$ , with Ricci curvature bounded below uniformly and volume non-collapsing, ( M k , g k ⁢ ( 0 ) ) ${(M_{k},g_{k}(0))}$ converge as a sequence of manifolds to a limit length space ( X , d ) ${(X,d)}$ . The regular part of ( X , d ) ${(X,d)}$ must be open and the convergence is of C 1 , α ${C^{1,\alpha}}$ regularity. As a corollary of this result, we prove that in dimension 3, a Ricci flow can always extend unless either its Ricci curvature goes to - ∞ ${-\infty}$ or it shrinks to a point.

Purpose With the increasing demand for surveillance and smart transportation, drone technology has become the center of attraction for robotics researchers. This study aims to introduce a new path planning approach to drone navigation based on topology in an uncertain environment. The main objective of this study is to use the Ricci flow evolution equation of metric and curvature tensor over angular Riemannian metric, and manifold for achieving navigational goals such as path length optimization at the minimum required time, collision-free obstacle avoidance in static and dynamic environments and reaching to the static and dynamic goals. The proposed navigational controller performs linearly and nonlinearly both with reduced error-based objective function by Riemannian metric and scalar curvature, respectively. Design/methodology/approach Topology and manifolds application-based methodology establishes the resultant drone. The trajectory planning and its optimization are controlled by the system of evolution equation over Ricci flow entropy. The navigation follows the Riemannian metric-based optimal path with an angular trajectory in the range from 0° to 360°. The obstacle avoidance in static and dynamic environments is controlled by the metric tensor and curvature tensor, respectively. The in-house drone is developed and coded using C++. For comparison of the real-time results and simulation results in static and dynamic environments, the simulation study has been conducted using MATLAB software. The proposed controller follows the topological programming constituted with manifold-based objective function and Riemannian metric, and scalar curvature-based constraints for linear and nonlinear navigation, respectively. Findings This proposed study demonstrates the possibility to develop the new topology-based efficient path planning approach for navigation of drone and provides a unique way to develop an innovative system having characteristics of static and dynamic obstacle avoidance and moving goal chasing in an uncertain environment. From the results obtained in the simulation and real-time environments, satisfactory agreements have been seen in terms of navigational parameters with the minimum error that justifies the significant working of the proposed controller. Additionally, the comparison of the proposed navigational controller with the other artificial intelligent controllers reveals performance improvement. Originality/value In this study, a new topological controller has been proposed for drone navigation. The topological drone navigation comprises the effective speed control and collision-free decisions corresponding to the Ricci flow equation and Ricci curvature over the Riemannian metric, respectively.

A smooth vector field ξ \xi on a Riemannian manifold ( M , g ) (M,g) is called a Ricci soliton if it satisfies the following Ricci soliton equation: \begin{equation*} \frac {1}{2}\pounds _{\xi }g+Ric=\lambda g, \end{equation*} where \pounds _{\xi }g denotes the Lie-derivative of the metric tensor g g with respect to ξ \xi , R i c Ric is the Ricci tensor and λ \lambda is a constant. Compact Ricci solitons are the fixed points of the Ricci flow projected from the space of metrics onto its quotient modulo diffeomorphisms and scalings, and often arise as blow-up limits for the Ricci flow on compact manifolds. Further, Ricci solitons model the formation of singularities in the Ricci flow and they correspond to self-similar solutions. In this survey article we present recent results on Ricci solitons which occur naturally on certain Riemannian submanifolds. In addition, we also present recent criteria of trivial compact shrinking Ricci solitons via Poisson’s equation.

Among the known examples of Einstein manifolds, a good many are Kahler. In fact, all compact examples with zero or negative scalar curvature are either Kahler, or locally homogeneous. On a complex manifold, one often gets Kahler-Einstein metrics by specific techniques. One reason is perhaps, in the Kahler case, the relative autonomy of the Ricci tensor with regard to the metric, once the complex structure is given. The Ricci tensor—or, to be precise, the Ricci form—only depends on the volume form. On the other hand, on a compact manifold, the cohomology class of the Ricci form is determined by the complex structure (so that, in particular, the sign of an eventual Kahler-Einstein metric is itself determined by the complex structure). Due to these circumstances, it has been possible to exhibit some existence theorems of Einstein metrics in the Kahler framework (Calabi-Yau and Aubin-Calabi-Yau theorems) which have no counterpart in general Riemannian geometry.

D'Atri and Nickerson [6], [7] have given necessary conditions for the geodesic symmetries of a Riemannian manifold to preserve the volume element. We use their results to show that ifG is a compact simple Lie group,T is a maximal torus ofG, andG/T is not symmetric, then anyG-invariant Kahler metric onG/T does not have volume-preserving geodesic symmetries. From the Kahler/de Rham decomposition of a compact homogeneous Kahler manifold [8], our result extends to the invariant Kahler metrics on a quotient of a compact connected Lie group by a maximal torus. In proving these results we compute directly the Ricci tensor of anyG-invariant Kahler metric onG/T forG compact connected andT a maximal torus ofG. The result is an explicit formula giving the value of the Ricci tensor elements in terms of the root structure of the Lie algebra ofG.

We study the behavior of the second order Renormalization Group flow on locally homogeneous metrics on closed three-manifolds. In the cases $\mathbb R^3$ and $\text{SO}(3)\times \R$, the flow is qualitatively the same as the Ricci flow. In the cases $\text{H}(3)$ and $\text{H}(2)\times \R$, if the curvature is small, then the flow expands as in the Ricci flow case, while if the curvature is large, then the flow contracts and forms a singularity in finite time. The main focus of the paper is the flow on the $\text{SU}(2)$, $\text{Nil}$, $\text{Sol}$, and $\text{SL}(2,\R)$ 3-geometries, with two of the three principal directions set equal. The configuration spaces for these geometries are two dimensional, and we can consequently apply phase plane techniques to the study. For the $\text{SU}(2)$ case, the flow is everywhere qualitatively the same as Ricci flow. For the $\text{Nil}$, $\text{Sol}$, and $\text{SL}(2,\R)$ cases, we show that the configuration space is partitioned into two regions which are delineated by a solution curve of the flow that depends on the coupling parameter: in one of the regions, the flow develops cigar or pancake singularities characteristic of the Ricci flow, while in the other both directions shrink. In the $\text{Nil}$ case we obtain a characterization of the full 3-dimensional flow.

The basic rigidity theorems for manifolds of nonnegative or positive Ricci curvature are the implies metric cone theorem, the maximal diameter theorem, [Cg], and the splitting theorem, [CG]. Each asserts that if a certain geometric quantity (volume or diameter) is as large as possible relative to the pertinent lower bound on Ricci curvature, then the metric on the manifold in question is a warped product metric of a particular type. In this paper we provide quantitative generalizations of the above mentioned results. Among the applications are the splitting theorem for GromovHausdorff limit spaces X, where Mn -* X, Ricmn ? -i see [FY]. Other applications include the assertion that for complete manifolds, M', with Ricmf > 0 and Euclidean volume growth, all tangent cones at infinity are metric cones; compare [BKN], [CT], [P1]. Via resealing arguments, there are also strong consequences for the local structure of manifolds whose Ricci curvature satisfies a fixed lower bound and for their Gromov-Hausdorff limits. Some of these are announced in [CCol]; for a more detailed discussion see [CCo2], [CCo3], [CCo4]. Our work further develops and significantly extends techniques which were introduced in [Col], [Co2] and significantly extended in [Co3], in order to prove certain stability conjectures of Anderson-Cheeger, Gromov and Perelman. The results of [Col]-[Co3] were announced in [Co4]. We briefly review some of those results. Let dGH denote the Gromov-Hausdorff distance between metric spaces; see [GLP]. Let S' denote the unit sphere and recall that S' is the unique complete

Let ( M n , g ) be a compact n-dimensional ( n ⩾ 2 ) manifold with nonnegative Ricci curvature, and if n ⩾ 3 , then we assume that ( M n , g ) × R has nonnegative isotropic curvature. The lower bound of the Ricci flow's existence time on ( M n , g ) is proved. This provides an alternative proof for the uniform lower bound of a family of closed Ricci flows' maximal existence times, which was first proved by E. Cabezas-Rivas and B. Wilking. We also get an interior curvature estimate for n = 3 under Rc ⩾ 0 assumption among others. Combining these results, we proved the short-time existence of the Ricci flow on a large class of three-dimensional open manifolds, which admit some suitable exhaustion covering and have nonnegative Ricci curvature.