Manifold Mn Research Articles

Extrinsic Geometry of a Riemannian Manifold and Ricci Solitons

The object of this paper is to find a vector field ξ and a constant λ on an n-dimensional compact Riemannian manifold Mn,g such that we obtain the Ricci soliton Mn,g,ξ,λ. In order to achieve this objective, we choose an isometric embedding provided in the work of Kuiper and Nash in the Euclidean space Rm,g¯ and choose ξ as the tangential component of a constant unit vector on Rm and call it a Kuiper–Nash vector. If τ is the scalar curvature of the compact Riemannian manifold Mn,g with a Kuiper–Nash vector ξ, we show that if the integral of the function ξτ has a suitable lower bound containing a constant λ, then Mn,g,ξ,λ is a Ricci soliton; we call this a Kuiper–Nash Ricci soliton. We find a necessary and sufficient condition involving the scalar curvature τ under which a compact Kuiper–Nash Ricci soliton Mn,g,ξ,λ is a trivial soliton. Finally, we find a characterization of an n-dimensional compact trivial Kuiper–Nash Ricci soliton Mn,g,ξ,λ using an upper bound on the integral of divξ2 containing the scalar curvature τ.

On the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds

We study the global topology of the space L of loops of contactomorphisms of a non-orderable closed contact manifold (M2n+1,ξ). We filter L by a quantitative measure of the “positivity” of the loops and describe the topology of L in terms of the subspaces of the filtration. In particular, we show that the homotopy groups of L are subgroups of the homotopy groups of the subspace of positive loops L+. We obtain analogous results for the space of loops of Legendrian submanifolds in (M2n+1,ξ).

Optimal coordinates for Ricci-flat conifolds

We compute the indicial roots of the Lichnerowicz Laplacian on Ricci-flat cones and give a detailed description of the corresponding radially homogeneous tensor fields in its kernel. For a Ricci-flat conifold (M, g) which may have asymptotically conical as well as conically singular ends, we compute at each end a lower bound for the order with which the metric converges to the tangent cone. As a special subcase of our result, we show that any Ricci-flat ALE manifold (Mn,g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(M^n,g)$$\\end{document} is of order n and thereby close a small gap in a paper by Cheeger and Tian.

On Morse–Smale diffeomorphisms on simply connected manifolds

We study relations between a structure of non-wandering set of a Morse–Smale diffeomorphism f and its carrying closed manifold Mn. We prove that if f has no any saddle periodic points with one-dimensional unstable manifolds, and for any periodic point σ of Morse index (n−1) its unstable manifolds do not intersect stable invariant manifolds of saddle periodic points different from σ, then Mn is simply connected. This fact does not follow from Morse inequalities that give only restrictions on homology groups of Mn.

A rigidity theorem for asymptotically flat static manifolds and its applications

In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds ( M n , g ) (M^{n},g) with boundary and with dimension n > 8 n>8 that was established by McCormick [Proc. Amer. Math. Soc. 146 (2018), pp. 4039–4046]. First, we show that any asymptotically flat static ( M n , g ) (M^{n},g) which achieves the equality and has CMC or equipotential boundary is isometric to a rotationally symmetric region of the Schwarzschild manifold. Then, we apply conformal techniques to derive a new Minkowski-type inequality for the level sets of bounded static potentials. Taken together, these provide a robust approach to detecting rotational symmetry of asymptotically flat static systems. As an application, we prove global uniqueness of static metric extensions for the Bartnik data induced by both Schwarzschild coordinate spheres and Euclidean coordinate spheres in dimension n > 8 n > 8 under the natural condition of Schwarzschild stability. This generalizes an earlier result of Miao [Classical Quantum Gravity 22 (2005), pp. L53–L59]. We also establish uniqueness for equipotential photon surfaces with small Einstein-Hilbert energy. This is interesting to compare with other recent uniqueness results for static photon surfaces and black holes, e.g. see V. Agostiniani and L. Mazzieri [Comm. Math. Phys. 355 (2017), pp. 261–301], C. Cederbaum and G. J. Galloway [J. Math. Phys. 62 (2021), p. 22], and S. Raulot [Classical Quantum Gravity 38 (2021), p. 22].

Rigidity properties of p-biharmonic maps and p-biharmonic submanifolds

We give some rigidity properties of a p-biharmonic map u:(M,g)→(N,h) between Riemannian manifolds (Mn,g) and (Nm,h). We first provide various sufficient conditions for p-biharmonic maps to be harmonic. Moreover, when the map u is an isometric immersion, by assuming that the Ln2-norm of the sectional curvature on M is sufficiently small or if the fundamental tone of the p-biharmonic submanifold is sufficiently big, it is proved that M is minimal.

The Yamabe flow on asymptotically flat manifolds

Abstract We study the Yamabe flow starting from an asymptotically flat manifold ( M n , g 0 ) (M^{n},g_{0}) . We show that the flow converges to an asymptotically flat, scalar flat metric in a weighted global sense if Y ⁢ ( M , [ g 0 ] ) > 0 Y(M,[g_{0}])>0 , and show that the flow does not converge otherwise. If the scalar curvature is nonnegative and integrable, then the ADM mass at time infinity drops by the limit of the total scalar curvature along the flow.

Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry

Let g0 be a smooth Riemannian metric on a closed manifold Mn of dimension n≥3. We study the existence of a smooth metric g conformal to g0 whose Schouten tensor Ag satisfies the differential inclusion λ(g−1Ag)∈Γ on Mn, where Γ⊂Rn is a cone satisfying standard assumptions. Inclusions of this type are often assumed in the existence theory for fully nonlinear elliptic equations in conformal geometry. We assume the existence of a continuous metric g1 conformal to g0 satisfying λ(g1−1Ag1)∈Γ′‾ in the viscosity sense on Mn, together with a nondegenerate ellipticity condition, where Γ′=Γ or Γ′ is a cone slightly smaller than Γ. In fact, we prove not only the existence of metrics satisfying such differential inclusions, but also existence and uniqueness results for fully nonlinear eigenvalue problems for the Schouten tensor. We also give a number of geometric applications of our results. We show that the sign of a nonlinear eigenvalue for the σ2 operator is a conformal invariant in three dimensions. We also give a generalisation of a theorem of Aubin & Ehrlick on pinching of the Ricci curvature, and an application in the study of Green's functions for fully nonlinear Yamabe problems.

Min-max theory for free boundary G-invariant minimal hypersurfaces

Given a compact Riemannian manifold Mn+1 with dimension 3≤n+1≤7 and ∂M≠∅, the free boundary min-max theory built by Li-Zhou [19] shows the existence of a smooth almost properly embedded minimal hypersurface with free boundary in M. In this paper, we generalize their constructions into equivariant settings. Specifically, let G be a compact Lie group acting as isometries on M with cohomogeneity at least 3. Then there exists a nontrivial smooth almost properly embedded free boundary G-invariant minimal hypersurface. Moreover, if the Ricci curvature of M is non-negative and ∂M is strictly convex, then there exist infinitely many properly embedded G-invariant minimal hypersurfaces with free boundary.

Open books and embeddings of smooth and contact manifolds

Abstract We discuss some embedding results in the category of open books, Lefschetz fibrations, contact manifolds and contact open books. First we prove an open book version of the Haefliger–Hirsch embedding theorem by showing that every k-connected closed n-manifold (n ≥ 7, k < (n − 4)/2) with signature zero admits an open book embedding in the trivial open book of 𝕊2n − k . We then prove that every closed manifold M 2n+1 that bounds an achiral Lefschetz fibration admits an open book embedding in the trivial open book of 𝕊2⌊3n/2⌋+3. We also prove that every closed manifold M 2n+1 bounding an achiral Lefschetz fibration admits a contact structure that isocontact embeds in the standard contact structure on ℝ2n+3. Finally, we give various examples of contact open book embeddings of contact (2n + 1)-manifolds in the trivial supporting open book of the standard contact structure on 𝕊4n+1.

On Strominger Kähler-like manifolds with degenerate torsion

In this paper, we study a special type of compact Hermitian manifolds that are Strominger Kähler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is Kähler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Kähler manifold. Previously, we have shown that any SKL manifold ( M n , g ) (M^n,g) is always pluriclosed, and when the manifold is compact and g g is not Kähler, it cannot admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when n = 2 n=2 , the SKL condition is equivalent to the Vaisman condition. In this paper, we give a classification for compact non-Kähler SKL manifolds in dimension 3 3 and those with degenerate torsion in higher dimensions. We also present some properties about SKL manifolds in general dimensions, for instance, given any compact non-Kähler SKL manifold, its Kähler form represents a non-trivial Aeppli cohomology class, the metric can never be locally conformal Kähler when n ≥ 3 n\geq 3 , and the manifold does not admit any Hermitian symplectic metric.

Hypersurfaces of metallic Riemannian manifolds as k-Almost Newton-Ricci solitons

This research investigates k-Almost Newton-Ricci solitons (k-ANRS) embedded in a metallic Riemannian manifold Mn having the potential function ?. Furthermore, we prove geodesic and minimal conditions for hypersurfaces of metallic Riemannian manifolds. Beside this, we have explained some applications of metallic Riemannian manifold admitting k-Almost Newton-Ricci solitons.

Existence of conformal metrics with prescribed Q-curvature on manifolds

We consider the problem of existence of conformal metrics with prescribed Q-curvature on riemannian n-dimensional manifolds (Mn,g0),5≤n≤7, not conformally diffeomorphic to the standard sphere Sn. Under the assumptions that Qg0 is semi-positive, Rg0 is non-negative and the prescribed function is flat near its critical points, we study the loss of compactness of the problem and we prove existence results through Euler-Hopf type criteria.

Generalized quasi-Einstein metrics and applications on generalized Robertson–Walker spacetimes

In this paper, we study generalized quasi-Einstein manifolds (Mn, g, V, λ) satisfying certain geometric conditions on its potential vector field V whenever it is harmonic, conformal, and parallel. First, we construct some integral formulas and obtain some triviality results. Then, we find some necessary conditions to construct a quasi-Einstein structure on (Mn, g, V, λ). Moreover, we prove that for any generalized Ricci soliton (M̄=I×fM,ḡ,ξ̄,λ), where ḡ is a generalized Robertson–Walker spacetime metric and the potential field ξ̄=h∂t+ξ is conformal, (M̄,ḡ) can be considered as the model of perfect fluids in general relativity. Moreover, the fiber (M, g) also satisfies the quasi-Einstein metric condition. Therefore, the state equation of (M̄=I×fM,ḡ) is presented. We also construct some explicit examples of generalized quasi-Einstein metrics by using a four-dimensional Walker metric.

Translating Solutions of the Nonparametric Mean Curvature Flow with Nonzero Neumann Boundary Data in Product Manifold Mn × ℝ

Translating Solutions of the Nonparametric Mean Curvature Flow with Nonzero Neumann Boundary Data in Product Manifold Mn × ℝ

Biharmonic maps and biharmonic submanifolds with small curvature integral

In this article, we study biharmonic maps and biharmonic submanifolds with small curvature integral. Let φ:(Mn,g)→(Nm,h) be a biharmonic map from a complete noncompact Riemannian manifold (Mn,g) into a Riemannian manifold (Nm,h) satisfying that the L2-norm of the tension field of the map is finite. If the domain manifold of the map satisfies a Sobolev inequality and the Ln2-norm of the sectional curvature on the image φ(M) is sufficiently small, then we are able to prove the harmonicity of the biharmonic map. It turns out that the fundamental tone of M is sufficiently large, then such a biharmonic map φ must be harmonic. In case where the map is an isometric immersion, we prove that if M satisfies a Sobolev inequality, then M must be minimal under the assumption that the Ln2-norm of the Ricci curvature on M is sufficiently small. Moreover it is shown that if the fundamental tone of a biharmonic submanifold is sufficiently big, then it is minimal.

Conformal Kaehler Euclidean submanifolds

Let f:M2n→R2n+ℓ, n≥5, be a conformal immersion into Euclidean space with codimension ℓ where M2n is a Kaehler manifold of complex dimension n free of points where all sectional curvatures vanish. For codimension ℓ=1 or ℓ=2 we show that at least locally such a submanifold can always be obtained in a rather simple way, namely, from an isometric immersion of the Kaehler manifold M2n into either R2n+1 or R2n+2, the latter being a class of submanifolds already extensively studied.

On a topological classification of multidimensional polar flows

The work solves the classification problem for structurally stable flows, which goes back to the classical works of Andronov, Pontryagin, Leontovich and Mayer. One of important examples of such flows is so-called Morse-Smale flow, whose non-wandering set consists of a finite number of fixed points and periodic trajectories. To date, there are exhaustive classification results for Morse-Smale flows given on manifolds whose dimension does not exceed three, and a very small number of results for higher dimensions. This is explained by increasing complexity of the topological problems that arise while describing the structure of the partition of a multidimensional phase space into trajectories. In this paper authors investigate the class G(Mn) of Morse-Smale flows on a closed connected orientable manifold Mn whose non-wandering set consists of exactly four points: a source, a sink, and two saddles. For the case when the dimension n of the supporting manifold is greater or equal than four, it is additionally assumed that one of the invariant manifolds for each saddle equilibrium state is one-dimensional. For flows from this class, authors describe the topology of the supporting manifold, estimate minimum number of heteroclinic curves, and obtain necessary and sufficient conditions of topological equivalence. Authors also describe an algorithm that constructs standard representative in each class of topological equivalence. One of the surprising results of this paper is that while for n=3 there is a countable set of manifolds that admit flows from class G(M3), there is only one supporting manifold (up to homeomorphism) for dimension n>3.

Geometrical and physical properties of W2-symmetric and -recurrent manifolds

The authors discuss mainly that the Riemannian manifold Mn admitting a unit preserving circle field ? in the present paper. A sufficient and necessary condition is given that Riemannian manifold Mn is an Einstein manifold by imposing some conditions on W2 curvature tensor. Further, this paper obtains the algebra representation of curvature tensors of a W2-recurrent Riemannian manifold Mn given by R???? = 1/d2 [d?d?R?? ? d?d?R?? + d?d?R?? ? d?d?R??].

Rigidity of Riemannian manifolds with vanishing generalized Bach tensor

A generalized Bach tensor Bijt with parameter t∈R is introduced. A Riemannian manifold (Mn,g) is called Bt-flat if its generalised Bach tensor Bijt≡0 for some parameter t. In this paper, we first study the rigidity of closed Bt-flat Riemannian manifolds with positive constant scalar curvature. When the dimension n=4, we prove that all Bt-flat manifolds that satisfy a point-wise inequality must be of positive constant sectional curvature. Similar rigidity results are also obtained in terms of the Yamabe invariant. Moreover, using the curvature estimates for the general dimension n≥4, we obtain an integral inequality for Bt-flat manifolds, and prove that the equality occurs if and only if these manifolds are of positive constant sectional curvature. In addition, we also obtain similar rigidity results for complete Bt-flat manifolds.