Complete Manifold Research Articles

Nonexistence and umbilicity of spacelike submanifolds with second fundamental form locally timelike

Under the assumption that the second fundamental form is locally timelike, we establish new nonexistence and umbilicity results concerning n-dimensional spacelike submanifolds immersed with parallel mean curvature vector in the (n+p)-dimensional de Sitter space mathbb {S}^{n+p}_q of index q, such that 1le qle p. Our approach is based on a Simon’s type inequality involving the norm of the total umbilicity tensor, obtained by Mariano in [17], jointly with suitable maximum principles due to Alías, Caminha and do Nascimento [6, 7] for complete noncompact Riemannian manifolds and a weak version of Omori–Yau’s maximum principle for stochastically complete Riemanian manifolds proved by Pigola, Rigoli and Setti [20, 21].

Short-Time Existence for Harmonic Map Heat Flow with Time-Dependent Metrics

In this work, we obtain a short time existence result for harmonic map heat flow coupled with a smooth family of complete metrics in the domain manifold. Our results generalize short time existence results for harmonic map heat flow by Li-Tam [The heat equation and harmonic maps of complete manifolds, Invent. Math., 1991] and Chen-Zhu [Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geometry, 2006]. In particular, we prove the short time existence of harmonic map heat flow along a complete Ricci flow $g(t)$ on $M$ into a complete manifold with curvature bounded from above with a smooth initial map of uniformly bounded energy density, under the assumptions that $|\text{Rm}(g(t))|\leq a/t$ and $g(t)$ is uniformly equivalent to $g(0)$.

Rigidity of Complete Manifolds with Weighted Poincaré Inequality

We consider complete Riemannian manifolds which satisfy a weighted Poincarè inequality and have the Ricci curvature bounded below in terms of the weight function. When the weight function has a nonzero limit at infinity, the structure of this class of manifolds at infinity is studied and certain splitting result is obtained. Our result can be viewed as an improvement of Li–Wang’s result in Li and Wang (Ann Sci École Norm Sup (4) 39(6):921–982, 2006. https://doi.org/10.1016/j.ansens.2006.11.001.

Geometry of positive scalar curvature on complete manifold

Abstract In this paper, we study the interplay of geometry and positive scalar curvature on a complete, non-compact manifold with non-negative Ricci curvature. On three-dimensional manifold, we prove a minimal volume growth, an estimate of integral of scalar curvature and width. On higher-dimensional manifold, we obtain a volume growth with a stronger condition.

High-fidelity multiqubit Rydberg gates via two-photon adiabatic rapid passage

We present a robust protocol for implementing high-fidelity multiqubit controlled phase gates (C k Z) on neutral atom qubits coupled to highly excited Rydberg states. Our approach is based on extending adiabatic rapid passage to two-photon excitation via a short-lived intermediate excited state common to alkali-atom Rydberg experiments, accounting for the full impact of spontaneous decay and differential AC Stark shifts from the complete manifold of hyperfine excited states. We evaluate and optimise gate performance, concluding that for Cs and currently available laser frequencies and powers, a CCZ gate with fidelity for three qubits and CCCZ with for four qubits is attainable in μs via this protocol. Higher fidelities are accessible with future technologies, and our results highlight the utility of neutral atom arrays for the native implementation of multiqubit unitaries.

Nonnegative solution of a class of double phase problems with logarithmic nonlinearity

This manuscript proves the existence of a nonnegative, nontrivial solution to a class of double-phase problems involving potential functions and logarithmic nonlinearity in the setting of Sobolev space on complete manifolds. Some applications are also being investigated. The arguments are based on the Nehari manifold and some variational techniques.

Dynamical renormalization and universality in classical multifield cosmological models

We study the scaling behavior of classical multifield cosmological models with complete scalar manifold (M,G) and positive smooth scalar potential Φ, introducing a dynamical renormalization group action which relates their UV and IR limits. We show that the RG flow of such models interpolates between a modification of the geodesic flow of (M,G) (obtained in the UV limit) and the gradient flow of (M,G,V) (obtained in the IR limit), where the classical effective potential V is proportional to 2Φ. Using this fact, we show that two-field models whose scalar manifold has constant Gaussian curvature equal to −1, 0 or 1 are infrared universal in the sense that they suffice to describe the first order IR approximants of cosmological orbits for all two-field models with positive smooth scalar potential.

Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds

Abstract Let ( M , g T ⁢ M ) {(M,g^{TM})} be a noncompact complete Riemannian manifold of dimension n, and let F ⊆ T ⁢ M {F\subseteq TM} be an integrable subbundle of TM. Let g F = g T ⁢ M | F {g^{F}=g^{TM}|_{F}} be the restricted metric on F and let k F {k^{F}} be the associated leafwise scalar curvature. Let f : M → S n ⁢ ( 1 ) {f:M\to S^{n}(1)} be a smooth area decreasing map along F, which is locally constant near infinity and of non-zero degree. We show that if k F > rk ⁢ ( F ) ⁢ ( rk ⁢ ( F ) - 1 ) {k^{F}>{\rm rk}(F)({\rm rk}(F)-1)} on the support of d ⁢ f {{\rm d}f} , and either TM or F is spin, then inf ⁡ ( k F ) < 0 {\inf(k^{F})<0} . As a consequence, we prove Gromov’s sharp foliated ⊗ ε {\otimes_{\varepsilon}} -twisting conjecture. Using the same method, we also extend two famous non-existence results due to Gromov and Lawson about Λ 2 {\Lambda^{2}} -enlargeable metrics (and/or manifolds) to the foliated case.

Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds

The notion of strictly outward minimising hull is investigated for open sets of finite perimeter sitting inside a complete noncompact Riemannian manifold. Under natural geometric assumptions on the ambient manifold, the strictly outward minimising hull Ω⁎ of a set Ω is characterised as a maximal volume solution of the least area problem with obstacle, where the obstacle is the set itself. In the case where Ω has C1,α-boundary, the area of ∂Ω⁎ is recovered as the limit of the p-capacities of Ω, as p→1+. Finally, building on the existence of strictly outward minimising exhaustions, a sharp isoperimetric inequality is deduced on complete noncompact manifolds with nonnegative Ricci curvature, provided 3≤n≤7.

Non-Self-Adjointness of the Klein–Gordon Operator on a Globally Hyperbolic and Geodesically Complete Manifold: An Example

We describe a Lorentzian manifold, that is globally hyperbolic and geodesically complete, but such that the (minimally coupled) Klein–Gordon operator with the standard domain is not essentially self-adjoint.

On the Geometry in the Large of Einstein-like Manifolds

Gray has presented the invariant orthogonal irreducible decomposition of the space of all covariant tensors of rank 3, obeying only the identities of the gradient of the Ricci tensor. This decomposition introduced the seven classes of Einstein-like manifolds, the Ricci tensors of which fulfill the defining condition of each subspace. The large-scale geometry of such manifolds has been studied by many geometers using the classical Bochner technique. However, the scope of this method is limited to compact Riemannian manifolds. In the present paper, we prove several Liouville-type theorems for certain classes of Einstein-like complete manifolds. This represents an illustration of the new possibilities of geometric analysis.

Existence Results for Double Phase Problem in Sobolev–Orlicz Spaces with Variable Exponents in Complete Manifold

In this paper, we study the existence of non-negative non-trivial solutions for a class of double-phase problems where the source term is a Caratheodory function that satisfies the Ambrosetti–Rabinowitz type condition in the framework of Sobolev–Orlicz spaces with variable exponents in complete manifold. Our approach is based on the Nehari manifold and some variational techniques. Furthermore, the Hölder ine-quality, continuous and compact embedding results are proved.

Quantum cohomology of symplectic flag manifolds

We compute the quantum cohomology of symplectic flag manifolds. Symplectic flag manifolds can be described by non-abelian GLSMs with superpotential. Although the ring relations cannot be directly read off from the equations of motion on the Coulomb branch due to complication introduced by the non-abelian gauge symmetry, it can be shown that they can be extracted from the localization formula in a gauge-invariant form. Our result is general for all symplectic flag manifolds, which reduces to previously established results on symplectic Grassmannians and complete symplectic flag manifolds derived by other means. We also explain why a (0, 2) deformation of the GLSM does not give rise to a deformation of the quantum cohomology.

Polynomial estimates for solutions of parametric elliptic equations on complete manifolds

"Let $P : \CI(M; E) \to \CI(M; F)$ be an order $\mu$ differential operator with coefficients $a$ and $P_k := P : H^{s_0 + k +\mu}(M; E) \to H^{s_0 + k}(M; F)$. We prove polynomial norm estimates for the solution $P_0^{-1}f$ of the form $$\|P_0^{-1}f\|_{H^{s_0 + k + \mu}(M; E)} \le C \sum_{q=0}^{k} \, \| P_0^{-1} \|^{q+1} \,\|a \|_{W^{|s_0|+k}}^{q} \, \| f \|_{H^{s_0 + k - q}},$$ (thus in higher order Sobolev spaces, which amounts also to a parametric regularity result). The assumptions are that $E, F \to M$ are Hermitian vector bundles and that $M$ is a complete manifold satisfying the Fr\'echet Finiteness Condition (FFC), which was introduced in (Kohr and Nistor, Annals of Global Analysis and Geometry, 2022). These estimates are useful for uncertainty quantification, since the coefficient $a$ can be regarded as a vector valued random variable. We use these results to prove integrability of the norm $\|P_k^{-1}f\|$ of the solution of $P_k u = f$ with respect to suitable Gaussian measures."

The derivatives of the heat kernel on complete manifolds

We present a new method of deriving estimates of the derivatives of the heat kernel on complete noncompact manifolds, based on an iteration argument. Applying these estimates we study the Lp-boundedness of Littlewood–Paley–Stein operators on a class of locally symmetric spaces.

On generalized quasi-Einstein manifolds

A rigidity result for a class of compact generalized quasi-Einstein manifolds with constant scalar curvature is obtained. Moreover, under some geometric assumptions, the rigidity for the non-compact case is also proved. Considering non constant scalar curvature, we characterize the generalized quasi-Einstein manifolds which is conformal to the Euclidean space and we show that there exist two classes of complete manifolds, which are obtained by considering potential functions and conformal factors either to be radial or invariant under the action of an (n-1) dimensional translation group. Explicit examples are given.

Local curvature estimates for the Ricci-harmonic flow

In this paper we give an explicit bound of Δg(t)u(t) and the local curvature estimates for the Ricci-harmonic flow ∂tg(t)=−2Ricg(t)+4du(t)⊗du(t),∂tu(t)=Δg(t)u(t)under the condition that the Ricci curvature is bounded along the flow. In the second part these local curvature estimates are extended to a class of generalized Ricci flow, introduced by the author Li (2019), whose stable points give Ricci-flat metrics on a complete manifold, and which is very close to the (K,N)-super Ricci flow recently defined by Li and Li (0000). Next we propose a conjecture for Einstein’s scalar field equations motivated by a result in the first part and the bounded L2-curvature conjecture recently solved by Klainerman et al. (2015). In the last part of this paper, we discuss the forward and backward uniqueness for the Ricci-harmonic flow.

From Harmonic Mappings to Ricci Flows Due to the Bochner Technique

The present paper is devoted to the study a global aspect of the geometry of harmonic mappings and, in particular, infinitesimal harmonic transformations, and represents the application of our results to the theory of Ricci solitons. These results will be obtained using the methods of Geometric analysis and, in particular, due to theorems of Yau, Li and Schoen on the connections between the geometry of a complete smooth manifold and the global behavior of its subharmonic functions.

On minimal kernels and Levi currents on weakly complete complex manifolds

A complex manifold X X is weakly complete if it admits a continuous plurisubharmonic exhaustion function ϕ \phi . The minimal kernels Σ X k , k ∈ [ 0 , ∞ ] \Sigma _X^k, k \in [0,\infty ] (the loci where all C k \mathcal {C}^k plurisubharmonic exhaustion functions fail to be strictly plurisubharmonic), introduced by Slodkowski-Tomassini, and the Levi currents, introduced by Sibony, are both concepts aimed at measuring how far X X is from being Stein. We compare these notions, prove that all Levi currents are supported by all the Σ X k \Sigma _X^k ’s, and give sufficient conditions for points in Σ X k \Sigma _X^k to be in the support of some Levi current. When X X is a surface and ϕ \phi can be chosen analytic, building on previous work by the second author, Slodkowski, and Tomassini, we prove the existence of a Levi current precisely supported on Σ X ∞ \Sigma _X^\infty , and give a classification of Levi currents on X X . In particular, unless X X is a modification of a Stein space, every point in X X is in the support of some Levi current.

Onpz–Laplacian System Involving Critical Nonlinearities

In this paper, we deal with the existence of at least two nonnegative nontrivial solutions to apz–Laplacian system involving critical nonlinearity in the context of Sobolev spaces with variable exponents on complete manifolds. We have established our main results by exploring both Nehari’s method and doing a refined analysis on the associated fiber map and some variational techniques.