Compact Manifold Research Articles

On a class of Hardy-Sobolev critical quasilinear elliptic systems on compact Riemannian manifolds

In this work, we investigate the existence and regularity of solutions to the critical system{−Δp,gu+a(x)|u|p−2u+b(x)[(p−1)|u|p−2+|v|p−2]pv=αp⋆(s)f(x)u|u|α−2|v|βdg(x,x0)sinM,−Δp,gv+b(x)[(p−1)|v|p−2+|u|p−2]pu+c(x)|v|p−2v=βp⋆(s)f(x)v|v|β−2|u|αdg(x,x0)sinM, where (M,g) is a smooth closed Riemannian manifold of dimension n≥2, dg is the Riemannian distance, Δp,g=divg(|∇gu|p−2∇gu) is the p-Laplace-Beltrami operator on (M,g), p∈(1,n), a,b,c∈C0,ϱ(M) for some ϱ∈(0,1), with b≡0 when 1<p<2, x0∈M, s∈[0,p), f is a smooth function in M with f(x0)=maxM⁡f>0, and α>1, β>1 are two real numbers such that α+β=p⋆(s), where p⋆(s)=p(n−s)/(n−p) denotes the critical Hardy-Sobolev exponent.

Existence of solutions to elliptic equations on compact Riemannian manifolds

The aim of this paper is to investigate the existence of weak solutions of a nonlinear elliptic problem with Dirichlet boundary value condition, in the framework of Sobolev spaces on compact Riemannian manifolds

A Classification of Compact Cohomogeneity One Locally Conformal Kähler Manifolds

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification GC of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case.

The five gradients inequality on differentiable manifolds

The goal of this paper is to derive the so-called five gradients inequality for optimal transport theory for general cost functions on two class of differentiable manifolds: locally compact Lie groups and compact Riemannian manifolds.

Spectral asymptotics for linear elasticity: the case of mixed boundary conditions

We establish two-term spectral asymptotics for the operator of linear elasticity with mixed boundary conditions on a smooth compact Riemannian manifold of arbitrary dimension. We illustrate our results by explicit examples in dimension two and three, thus verifying our general formulae both analytically and numerically.

Brownian motion in the Hilbert space of quantum states and the stochastically emergent Lorentz symmetry: A fractal geometric approach from Wiener process to formulating Feynman’s path-integral measure for relativistic quantum fields

This paper aims to provide a consistent, finite-valued, and mathematically well-defined reformulation of Feynman’s path-integral measure for quantum fields obtained by studying the Wiener stochastic process in the infinite-dimensional Hilbert space of quantum states. This reformulation will undoubtedly have a crucial role in formulating quantum gravity within a mathematically well-defined framework. In fact, this study is fundamentally different from any previous research on the relationship between Feynman’s path-integral and the Wiener stochastic process. In this research, we focus on the fact that the classic Wiener measure is no longer applicable in infinite-dimensional Hilbert spaces due to fundamental differences between displacements in low and extremely high dimensions. Thus, an analytic norm motivated by the role of the fractal functions in the Wilsonian renormalization approach is worked out to properly characterize Brownian motion in the Hilbert space of quantum states on a compact flat manifold. This norm, the so-called fractal norm, pushes the rougher functions (physically the quantum states with higher energies) to the farther points of the Hilbert space until the fractal functions as the roughest ones are moved to infinity. Implementing the Wiener stochastic process with the fractal norm, results in a modified form of the Wiener measure called the Wiener fractal measure, which is a well-defined measure for Feynman’s path-integral formulation of quantum fields. Wiener fractal measure has a complicated formula of non-local terms but produces the Klein–Gordon action at the first order of approximation. Using complex integrals to compensate for the removal of non-local terms appearing in higher orders of approximation, the Wiener fractal measure turns into a complex measure and generates Feynman’s path-integral formulation of scalar quantum fields. This brings us to the main objective of this study. Finally, some various significant aspects of quantum field theory (such as renormalizability, RG flow, Wick rotation, regularization, etc.) are revisited by means of the analytical aspects of the Wiener fractal measure.

A note on the long neck principle and spectral width inequality of geodesic collar neighborhoods

The main purpose of this short note is to derive some generalizations of the long neck principle and give a spectral width inequality of geodesic collar neighborhoods. Our results are obtained via the spinorial Callias operator approach. An important step is to introduce the relative Gromov-Lawson pair on a compact manifold with boundary, relative to a background manifold.

Almost sure scattering for the defocusing cubic nonlinear Schrödinger equation on [formula omitted]

We consider the Cauchy problem for the defocusing cubic nonlinear Schrödinger equation (NLS) on the waveguide manifold R3×T and establish almost sure scattering for random initial data, where no symmetry conditions are imposed and the result is available for arbitrarily rough data f∈Hs with s∈R. The main new ingredient is a layer-by-layer refinement of the newly established randomization introduced by Shen-Soffer-Wu [40], which enables us to also obtain a strongly smoothing effect from the randomization for the forcing term along the periodic direction. It is worth noting that such a smoothing effect generally can not hold for purely compact manifolds, which is on the contrary available for the present model thanks to the mixed type nature of the underlying domain. As a byproduct, by assuming that the initial data are periodically trivial, we also obtain almost sure scattering for the defocusing cubic NLS on R3 which parallels the result by Camps [15] and Shen-Soffer-Wu [41]. To our knowledge, the paper also gives the first almost sure well-posedness and scattering result for NLS on product spaces.

Universal fine grained asymptotics of free and weakly coupled quantum field theory

We give a rigorous proof that in any free quantum field theory with a finite group global symmetry G, on a compact spatial manifold, at sufficiently high energy, the density of states ρα(E) for each irreducible representation α of G obeys a universal formula as conjectured by Harlow and Ooguri. We further prove that this continues to hold in a weakly coupled quantum field theory, given an appropriate scaling of the coupling with temperature. This generalizes similar results that were previously obtained in (1 + 1)-D to higher spacetime dimension. We discuss the role of averaging in the density of states, and we compare and contrast with the case of continuous group G, where we prove a universal, albeit different, behavior.

A remark on inverse problems for nonlinear magnetic Schrödinger equations on complex manifolds

We show that the knowledge of the Dirichlet–to–Neumann map for a nonlinear magnetic Schrödinger operator on the boundary of a compact complex manifold, equipped with a Kähler metric and admitting sufficiently many global holomorphic functions, determines the nonlinear magnetic and electric potentials uniquely.

Automorphisms of projective manifolds

Let $(M,P\nabla_M)$ be a compact projective manifold and $Aut(M,P\nabla_M)$ its group of automorphisms. The purpose of this paper is to study the topological properties of $(M,P\nabla_M)$ if $Aut(M,P\nabla_M))$ is not discrete by applying the results of [13] and the Benzekri's functor which associates to a projective manifold a radiant affine manifold. This enables us to show that the orbits of the connected component of $Aut(M,P\nabla_M)$ are immersed projective submanifolds. We also classify $3$-dimensional compact projective manifolds such that $dim(Aut(M,P\nabla_M))\geq 2$.

A Maximal Oscillatory Operator on Compact Manifolds

A Maximal Oscillatory Operator on Compact Manifolds

Kähler-Yang-Mills Equations and Vortices

The Kähler-Yang-Mills equations are coupled equations for a Kähler metric on a compact complex manifold and a connection on a complex vector bundle over it. After briefly reviewing the main aspects of the geometry of the Kähler-Yang-Mills equations, we consider dimensional reductions of the equations related to vortices — solutions to certain Yang-Mills-Higgs equations.

Advancements in the Analysis of Sobolev Spaces and Function Spaces on Manifolds: Theoretical Framework and Applications

This research paper delves into Sobolev spaces and function spaces on smooth manifolds, revealing fundamental theorems such as existence, embeddings, and compactness properties. Noteworthy results include the Poincare inequality elucidating function behavior on compact manifolds and compactness properties of Sobolev spaces on Riemannian manifolds. The study establishes trace theorems for functions on the boundary and interpolation results between Sobolev spaces. Isoperimetric inequalities and stability under weak convergence contribute to a holistic understanding of geometric and analytical aspects of Sobolev spaces. The research concludes by exploring invariance under diffeomorphisms and compactness in dual spaces, providing a unified framework for analyzing function spaces on manifolds.

HKT Manifolds: Hodge Theory, Formality and Balanced Metrics

ABSTRACT Let $(M,I,J,K,\Omega)$ be a compact HKT manifold, and let us denote with $\partial$ the conjugate Dolbeault operator with respect to I, $\partial_J:=J^{-1}\overline\partial J$, $\partial^\Lambda:=[\partial,\Lambda]$, where Λ is the adjoint of $L:=\Omega\wedge-$. Under suitable assumptions, we study Hodge theory for the complexes $(A^{\bullet,0},\partial,\partial_J)$ and $(A^{\bullet,0},\partial,\partial^\Lambda)$ showing a similar behavior to Kähler manifolds. In particular, several relations among the Laplacians, the spaces of harmonic forms and the associated cohomology groups, together with Hard Lefschetz properties, are proved. Moreover, we show that for a compact HKT $\mathrm{SL}(n,\mathbb{H})$-manifold, the differential graded algebra $(A^{\bullet,0},\partial)$ is formal and this will lead to an obstruction for the existence of an HKT $\mathrm{SL}(n,\mathbb{H})$ structure $(I,J,K,\Omega)$ on a compact complex manifold (M, I). Finally, balanced HKT structures on solvmanifolds are studied.

Almost Sure Scattering for the One Dimensional Nonlinear Schrödinger Equation

We consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree p &gt; 1 p&gt;1 . On compact manifolds many probability measures are invariant by the flow of the linear Schrödinger equation (e.g. Wiener measures), and it is sometimes possible to modify them suitably and get invariant (Gibbs measures) or quasi-invariant measures for the non linear problem. On R d \mathbb {R}^d , the large time dispersion shows that the only invariant measure is the δ \delta measure on the trivial solution u = 0 u =0 , and the good notion to track is whether the non linear evolution of the initial measure is well described by the linear (nontrivial) evolution. This is precisely what we achieve in this work. We exhibit measures on the space of initial data for which we describe the nontrivial evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. Actually, we give precise (and optimal) bounds on the Radon–Nikodym derivatives of these measures with respect to each other and we characterise their L p L^p regularity. We deduce from this precise description the global well-posedness of the equation for p &gt; 1 p&gt;1 and scattering for p &gt; 3 p&gt;3 (actually even for 1 &gt; p ≤ 3 1&gt;p \leq 3 , we get a dispersive property of the solutions and exhibit an almost sure polynomial decay in time of their L p + 1 L^{p+1} norm). To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties or decay) for the nonlinear evolution.

Monge-Ampère equations on compact Hessian manifolds

We consider degenerate Monge-Ampere equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Ampere operators. We then use the Perron method to solve Monge-Ampere equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delanoe, Caffarelli-Viaclovsky and Hultgren-Onnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow conjecture.

On arithmetic quotients of the group SL2 over a quaternion division k-algebra

Abstract Given a totally real algebraic number field k of degree s, we consider locally symmetric spaces X G / Γ {X_{G}/\Gamma} associated with arithmetic subgroups Γ of the special linear algebraic k-group G = SL M 2 ⁢ ( D ) {G=\mathrm{SL}_{M_{2}(D)}} , attached to a quaternion division k-algebra D. The group G is k-simple, of k-rank one, and non-split over k. Using reduction theory, one can construct an open subset Y Γ ⊂ X G / Γ {Y_{\Gamma}\subset X_{G}/\Gamma} such that its closure Y ¯ Γ {\overline{Y}_{\Gamma}} is a compact manifold with boundary ∂ ⁡ Y ¯ Γ {\partial\overline{Y}_{\Gamma}} , and the inclusion Y ¯ Γ → X G / Γ {\overline{Y}_{\Gamma}\rightarrow X_{G}/\Gamma} is a homotopy equivalence. The connected components Y [ P ] {Y^{[P]}} of the boundary ∂ ⁡ Y ¯ Γ {\partial\overline{Y}_{\Gamma}} are in one-to-one correspondence with the finite set of Γ-conjugacy classes of minimal parabolic k-subgroups of G. We show that each boundary component carries the natural structure of a torus bundle. Firstly, if the quaternion division k-algebra D is totally definite, that is, D ramifies at all archimedean places of k, we prove that the basis of this bundle is homeomorphic to the torus T s - 1 {T^{s-1}} of dimension s - 1 {s-1} , has the compact fibre T 4 ⁢ s {T^{4s}} , and its structure group is SL 4 ⁢ s ⁢ ( ℤ ) {\mathrm{SL}_{4s}(\mathbb{Z})} . We determine the cohomology of Y [ P ] {Y^{[P]}} . Secondly, if the quaternion division k-algebra D is indefinite, thus, there exists at least one archimedean place v ∈ V k , ∞ {v\in V_{k,\infty}} at which D v {D_{v}} splits over ℝ {\mathbb{R}} , that is, D v ≅ M 2 ⁢ ( ℝ ) {D_{v}\cong M_{2}(\mathbb{R})} , the fibre is homeomorphic to T 4 ⁢ s {T^{4s}} , but the base space of the bundle is more complicated.

A new approach from the calculus of variations to the Lichnerowicz theorem

The well-known Lichnerowicz theorem states that on an n-dimensional compact Riemannian manifold without boundary with the assumptionRicg≥(n−1)g on the Ricci curvature, the first eigenvalue λ of the Laplacian on the manifold satisfiesλ≥n. This theorem is proved using a Bochner formula.In this paper we give a new approach from the calculus of variations to the Lichnerowicz theorem. We use the energy Econf() of conformality, introduced in [3]. For a smooth map f between Riemannian manifolds, the map f is a weakly conformal map (almost everywhere) if and only if Econf(f) = 0 (the minimum value).Our approach is as follows:(1)The identity map on M is a conformal map, hence a minimizer of the energy Econf(). Especially the identity map is stable, i.e., the second variation is non-negative for the identity map.(2)For the first eigenfunction φ, we take the variation vector field corresponding to dφ, and the non-negativity of the second variation for the variation vector field at the identity map implies the Lichnerowicz theorem.

Cherrier–Escobar problem for the elliptic Schrödinger-to-Neumann map

In this paper, we study a Cherrier–Escobar problem for the extended problem corresponding to the elliptic Schrödinger-to-Neumann map on a compact 3-dimensional Riemannian manifold with boundary. Using the algebraic topological argument of Bahri and Coron (1988), we show solvability under the assumption that the extended problem corresponding to the elliptic Schrödinger-to-Neumann map has a positive first eigenvalue and a positive Green’s function.