Complete Manifold Research Articles

A critical point analysis of Landau–Ginzburg potentials with bulk in Gelfand–Cetlin systems

Using the bulk deformation of Floer cohomology by Schubert classes and non-Archimedean analysis of Fukaya–Oh–Ohta–Ono’s bulk-deformed potential function, we prove that every complete flag manifold Fl(n) (n≥3) with a monotone Kirillov–Kostant–Souriau (KKS) symplectic form carries a continuum of nondisplaceable Lagrangian tori which degenerates to a nontorus fiber in the Hausdorff limit. In particular, the Lagrangian S3-fiber in Fl(3) is nondisplaceable, answering a question raised by Nohara and Ueda who computed its Floer cohomology to be vanishing.

Geodesic neighborhoods in unitary orbits of self-adjoint operators of [formula omitted

In the present paper, we study the unitary orbit of a compact Hermitian diagonal operator with spectral multiplicity one under the action of the unitary group UK+C of the unitization of the compact operators K(H)+C, or equivalently, the quotient UK+C/UD(K+C). We relate this and the action of different unitary subgroups to describe metric geodesics (using a natural distance) which join end points. As a consequence we obtain a local Hopf-Rinow theorem. We also explore cases about the uniqueness of short curves and prove that there exist some of these that cannot be parameterized using minimal anti-Hermitian operators of K(H)+C.

Slice Rigidity Property of Holomorphic Maps Kobayashi-Isometrically Preserving Complex Geodesics

In this paper, we study the following “slice rigidity property”: given two Kobayashi complete hyperbolic manifolds M, N and a collection of complex geodesics \({\mathscr {F}}\) of M, when is it true that every holomorphic map \(F:M\rightarrow N\) which maps isometrically every complex geodesic of \({\mathscr {F}}\) onto a complex geodesic of N is a biholomorphism? Among other things, we prove that this is the case if M, N are smooth bounded strictly (linearly) convex domains, every element of \({\mathscr {F}}\) contains a given point of \({\overline{M}}\) and \({\mathscr {F}}\) spans all of M. More general results are provided in dimension 2 and for the unit ball.

Heterotic line bundle models on generalized complete intersection Calabi Yau manifolds

The systematic program of heterotic line bundle model building has resulted in a wealth of standard-like models (SLM) for particle physics. In this paper, we continue this work in the setting of generalised Complete Intersection Calabi Yau (gCICY) manifolds. Using the gCICYs constructed in ref. [1], we identify two geometries that, when combined with line bundle sums, are directly suitable for heterotic GUT models. We then show that these gCICYs admit freely acting ℤ2 symmetry groups, and are thus amenable to Wilson line breaking of the GUT gauge group to that of the standard model. We proceed to a systematic scan over line bundle sums over these geometries, that result in 99 and 33 SLMs, respectively. For the first class of models, our results may be compared to line bundle models on homotopically equivalent Complete Intersection Calabi Yau manifolds. This shows that the number of realistic configurations is of the same order of magnitude.

Analytic structure of all loop banana integrals

Using the Gelfand-Kapranov-Zelevinskĭ system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana integrals with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel hat{Gamma}hbox{-} mathrm{class} evaluation in the ambient spaces of the mirror, while the imaginary part of the integral in this regime is determined by the hat{Gamma}hbox{-} mathrm{class} of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius κ-constants, which determine the behaviour of the integrals when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogeneous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the banana integral in very short time to very high numerical precision for all values of the physical parameters. Using modular properties of the periods we determine the value of the maximal cut equal mass four-loop integral at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins.

On differential equations classifying a warped product submanifold of cosymplectic space forms

In the present paper, we extend the study of (Ali et al. in J. Inequal. Appl. 2020:241, 2020) by using differential equations (García-Río et al. in J. Differ. Equ. 194(2):287–299, 2003; Pigola et al. in Math. Z. 268:777–790, 2011; Tanno in J. Math. Soc. Jpn. 30(3):509–531, 1978; Tashiro in Trans. Am. Math. Soc. 117:251–275, 1965), and we find some necessary conditions for the base of warped product submanifolds of cosymplectic space form widetilde{M}^{2m+1}(epsilon ) to be isometric to the Euclidean space mathbb{R}^{n} or a warped product of complete manifold N and Euclidean space mathbb{R}.

Heat kernel upper bounds for symmetric Markov semigroups

It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this paper, we show that both imply (and hence are equivalent to) off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. Our results extend that of [6] for Nash-type inequalities with power order considerably and also extend that of [26] for second order differential operators on a complete non-compact manifold.

Fano deformation rigidity of rational homogeneous spaces of submaximal Picard numbers

We study the question whether rational homogeneous spaces are rigid under Fano deformation. In other words, given any smooth connected family \(\pi :{{\mathcal {X}}}\rightarrow {{\mathcal {Z}}}\) of Fano manifolds, if one fiber is biholomorphic to a rational homogeneous space S, is \(\pi \) an S-fibration? The cases of Picard number one were answered by Hwang and Mok. The manifold \({{\mathbb {F}}}(1, Q^5)\) is the unique rational homogeneous space of Picard number one that is not rigid under Fano deformation, and a Fano degeneration of it is constructed by Pasquier and Perrin. For higher Picard number cases, one notices that the Picard number of a rational homogeneous space G/P satisfies \(\rho (G/P)\le \mathrm{rank}(G)\). Weber and Wiśniewski proved that the rational homogeneous spaces G/P with \(\rho (G/P)=\mathrm{rank}(G)\) (i.e. complete flag manifolds) are rigid under Fano deformation. In this paper, we show that the rational homogeneous spaces G/P with \(\rho (G/P)=\mathrm{rank}(G)-1\) are rigid under Fano deformation, provided that G is a simple algebraic group of type ADE, and G/P is not biholomorphic to \({{\mathbb {F}}}(1, 2, {{\mathbb {P}}}^3)\) or \({{\mathbb {F}}}(1, 2, Q^6)\). We also show that \({{\mathbb {F}}}(1, 2, {{\mathbb {P}}}^3)\) has a unique Fano degeneration, which is explicitly constructed. Furthermore, the structure of possible Fano degenerations of \({{\mathbb {F}}}(1, 2, Q^6)\) is also described explicitly. Our main result is obtained by applying the theory of Cartan connections and symbol algebras.

Weakly PIC1 Manifolds with Maximal Volume Growth

In this article we study the local estimates of Ricci flow when the curvature of the initial metric is of almost PIC1 and use it to study complete manifolds with weakly PIC1 and maximal volume growth.

Sufficient Criteria for Obtaining Hardy Inequalities on Finsler Manifolds

We establish Hardy inequalities involving a weight function on complete, not necessarily reversible Finsler manifolds. We prove that the superharmonicity of the weight function provides a sufficient condition to obtain Hardy inequalities. Namely, if \(\rho \) is a nonnegative function and \(-\varvec{\Delta } \rho \ge 0\) in weak sense, where \(\varvec{\Delta }\) is the Finsler–Laplace operator defined by \( \varvec{\Delta } \rho = \mathrm {div}(\varvec{\nabla } \rho )\), then we obtain the generalization of some Riemannian Hardy inequalities given in D’Ambrosio and Dipierro (Ann Inst H Poincaré Anal Non Linéaire 31(3):449–475, 2014). By extending the results obtained, we prove a weighted Caccioppoli-type inequality, a Gagliardo–Nirenberg inequality and a Heisenberg–Pauli–Weyl uncertainty principle on complete Finsler manifolds. Furthermore, we present some Hardy inequalities on Finsler–Hadamard manifolds with finite reversibility constant, by defining the weight function with the help of the distance function. Finally, we extend a weighted Hardy inequality to a class of Finsler manifolds of bounded geometry.

Complete noncompact Spin(7) manifolds from self-dual Einstein 4–orbifolds

We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2 orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of 4-dimensional ALF hyperk\"ahler metrics. We apply our construction to asymptotically conical G2 metrics arising from self-dual Einstein 4-orbifolds with positive scalar curvature. As illustrative examples of the power of our construction, we produce complete non-compact Spin(7) manifolds with arbitrarily large second Betti number and infinitely many distinct families of ALC Spin(7) metrics on the same smooth 8-manifold.

Periodic solutions to Navier-Stokes equations on non-compact Einstein manifolds with negative curvature

Consider the Navier-Stokes Equations (NSE) for viscous incompressible fluid flows on a non-compact, smooth, simply-connected and complete Einstein manifold $$(\mathbf {M},g)$$ with negative Ricci curvature tensor. We prove the existence and uniqueness of a time-periodic solution to NSE for vector fields on $$(\mathbf {M},g)$$ . Our method is based on the dispersive and smoothing properties of the semigroup generated by the linearized Stokes equations to construct a bounded (in time) solution of the nonhomogeneous Stokes equation and on the ergodic method to obtain the periodic solution to Stokes equation. Then, using the fixed point arguments, we can pass from the Stokes equations to Navier-Stokes equations to obtain periodic solutions to NSE on the Einstein manifold $$(\mathbf {M},g)$$ . We also prove the stability of the periodic solution.

Affine functions and Busemann functions on complete Finsler manifolds

We clarify the relation between an affine function and a Busemann function in a geodesically complete Finsler manifold. As an application, we give the characterization of a Minkowski space by means of the dimension of the vector space consisting of all affine functions on a Finsler manifold.

The $k$-almost Yamabe solitons and contact metric manifolds

We introduce the concept of a k-almost Yamabe soliton which extends naturally from Yamabe solitons. Our aim is to study the k-almost Yamabe soliton (g,V,k,λ) on a contact metric manifold M2n+1. Firstly, for a general contact metric manifold, it is proved that V is Killing if the potential vector field V is a contact vector field and that M is K-contact if V is collinear with Reeb vector field. Secondly, we prove that a compact K-contact manifold, admitting a k-almost Yamabe gradient soliton, is isometric to a standard unit sphere. Moreover, for a complete Sasakian manifold admitting a k-almost Yamabe soliton, we show that it is isometric to a standard unit sphere 𝕊2n+1(1) when n>1 and for n=1, M is also isometric to a standard unit sphere if it admits a closed k-almost Yamabe soliton. Finally, we consider a contact metric (κ,μ)-manifold with a nontrivial k-almost Yamabe gradient soliton and show that it is flat in dimension 3 and in higher dimension M is locally isometric to En+1×𝕊n(4). In the end, we construct two examples of contact metric manifolds with a k-almost Yamabe soliton.

Solutions of the $$\bar{\partial }$$-equation with compact support on Stein and Kähler manifold

We study the $$\bar{\partial }$$ -equation first in Stein manifold then in complete Kahler manifolds. The aim is to get $$L^{r}$$ and Sobolev estimates on solutions with compact support. In the Stein case we get that for any (p, q)-form $$\omega$$ in $$L^{r}$$ with compact support and $$\bar{\partial }$$ -closed there is a $$(p,q-1)$$ -form u in $$W^{1,r}$$ with compact support and such that $$\bar{\partial } u=\omega .$$ In the case of Kahler manifold, we prove and use estimates on solutions on Poisson equation with compact support and the link with $$\bar{\partial }$$ equation is done by a classical theorem stating that the Hodge Laplacian is twice the $$\bar{\partial }$$ (or Kohn) Laplacian in a Kahler manifold. This uses and improves, in special cases, our result on Andreotti–Grauert-type theorem.

The base of warped product submanifolds of Sasakian space forms characterized by differential equations

In the present paper, we find some characterization theorems. Under certain pinching conditions on the warping function satisfying some differential equation, we show that the base of warped product submanifolds of a Sasakian space form widetilde{M}^{2m+1}(epsilon ) is isometric either to a Euclidean space mathbb{R}^{n} or a warped product of a complete manifold N and the Euclidean line mathbb{R}.

Gradient Estimate and Liouville Theorems for p-Harmonic Maps

In this paper, we first obtain an $$L^q$$ L q gradient estimate for p-harmonic maps, by assuming the target manifold supporting a certain function, whose gradient and Hessian satisfy some analysis conditions. From this $$L^q$$ L q gradient estimate, we get a corresponding Liouville type result for p-harmonic maps. Secondly, using these general results, we give various geometric applications to p-harmonic maps from complete manifolds with nonnegative Ricci curvature to manifolds with various upper bound on sectional curvature, under appropriate controlled images.

Local curvature estimates for the Laplacian flow

In this paper we give local curvature estimates for the Laplacian flow on closed G_{2}-structures under the condition that the Ricci curvature is bounded along the flow. The main ingredient consists of the idea of Kotschwar et al. (J Funct Anal 271(9):2604–2630, 2016) who gave local curvature estimates for the Ricci flow on complete manifolds and then provided a new elementary proof of Sesum’s result (Sesum in Am J Math 127(6):1315–1324, 2005), and the particular structure of the Laplacian flow on closed G_{2}-structures. As an immediate consequence, this estimates give a new proof of Lotay and Wei’s (Geom Funct Anal 27(1):165–233, 2017) result which is an analogue of Sesum’s theorem. The second result is about an interesting evolution equation for the scalar curvature of the Laplacian flow of closed G_{2}-structures. Roughly speaking, we can prove that the time derivative of the scalar curvature R_{g(t)} is equal to the Laplacian of R_{g(t)}, plus an extra term which can be written as the difference of two nonnegative quantities.

L2 vanishing theorem on some Kähler manifolds

Let E be a Hermitian vector bundle over a complete Kahler manifold (X, ω), dimℂX = n, with a d(bounded) Kahler form ω, and let dA be a Hermitian connection on E. The goal of this article is to study the L2-Hodge theory on the vector bundle E. We extend the results of Gromov [18] to the Hermitian vector bundle. Finally, as an application, we prove a gap result for the Yang-Mills connection on the bundle E over X.

Stationary–Complete Spacetimes with non-standard splittings and pre-Randers metrics

Using the relativistic Fermat’s principle, we establish a bridge between stationary–complete manifolds which satisfy the observer-manifold condition and pre-Randers metrics, namely, Randers metrics without any restriction on the one-form. As a consequence, we give a description of the causal ladder of such spacetimes in terms of the elements associated with the pre-Randers metric: its geodesics and the associated distance. We obtain, as applications of this interplay, the description of conformal maps of Killing submersions, and existence and multiplicity results for geodesics of pre-Randers metrics and magnetic geodesics.