Orientable Riemannian Manifold Research Articles

On partial Hölder continuity and a Caccioppoli inequality for minimizers of asymptotically convex functionals between Riemannian manifolds

We initiate a low-order regularity theory for vectorial minimizers of the functional Open image in new window where \(\varvec{\textit{w}}:\mathcal {M}\rightarrow \mathcal {N}\) and \(\mathcal {M}\) and \(\mathcal {N}\) are orientable Riemannian manifolds, in the case where G is only asymptotically convex. In particular, we prove three interrelated regularity results. The first result establishes partial Morrey regularity estimates, and Holder continuity, for minimizers. For the second result by utilizing the obtained partial Holder continuity, we prove a partial Caccioppoli-type inequality for minimizers. Finally, this result then allows us to deduce a higher integrability result for the gradient of minimizers of this functional. Since we work in the asymptotically convex setting, our results apply to far more general integrands than previously available in the literature for this type of problem.

Calabi-Yau structures on cotangent bundles

Starting with a orientable compact real-analytic Riemannian manifold $(L,g)$ with $\chi (L) = 0$, we show that a small neighborhood $\mathrm{Op}(L)$ of the zero section in the cotangent bundle $T*L$ carries a Calabi–Yau structure such that the zero section is an isometrically embedded special Lagrangian submanifold.

Holomorphic harmonic morphisms from cosymplectic almost Hermitian manifolds

We study 4-dimensional orientable Riemannian manifolds equipped with a minimal and conformal foliation $${\mathcal {F}}$$ of codimension 2. We prove that the two adapted almost Hermitian structures $$J_1$$ and $$J_2$$ are both cosymplectic if and only if $${\mathcal {F}}$$ is Riemannian and its horizontal distribution $${\mathcal {H}}$$ is integrable.

Stability and integrability of horizontally conformal maps and harmonic morphisms

In this article, we study stability of minimal fibers and integrability of horizontal distribution for horizontally conformal maps and harmonic morphisms. Let be a horizontally conformal submersion. We prove that if the horizontal distribution is integrable, then any minimal fiber of φ is volume‐stable. This result is an improved version of the main theorem in [15]. As a corollary, we obtain if φ is a submersive harmonic morphism whose fibers are totally geodesic, and the horizontal distribution is integrable, then any fiber of φ is volume‐stable and so such a map φ is energy‐stable if M is compact. We also show that if is a horizontally conformal map from a compact Riemannian manifold M into an orientable Riemannian manifold N which is horizontally homothetic, and if the pull‐back of the volume form of N is harmonic, then the horizontal distribution is integrable and φ is a harmonic morphism.

Real Structures on Almost-Commutative Spectral Triples

We refine the reconstruction theorem for almost-commutative spectral triples to a result for real almost-commutative spectral triples, clarifying, in the process, both concrete and abstract definitions of real commutative and almost-commutative spectral triples. In particular, we find that a real almost-commutative spectral triple algebraically encodes the commutative *-algebra of the base manifold in a canonical way, and that a compact oriented Riemannian manifold admits real (almost-)commutative spectral triples of arbitrary KO-dimension. Moreover, we define a notion of smooth family of real finite spectral triples and of the twisting of a concrete real commutative spectral triple by such a family, with interesting KK-theoretic and gauge-theoretic implications.

The η invariant of the Atiyah–Patodi–Singer operator on compact flat manifolds

Let \({\mathcal{D}}\) be the boundary operator defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact orientable Riemannian manifold M. In continuation of our previous study, we deal with the problem of computing explicitly the η invariant η = η(M) for any orientable compact flat manifold M. After giving an explicit expression for η(s) in the case of cyclic holonomy group, we obtain a combinatorial formula that reduces the computation to the cyclic case. We illustrate the method by determining η(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p ≥ 7, η(s) can be expressed as a multiple of Lχ(s), an L-function associated to a quadratic character mod p, while η(0) is a (non-zero) integral multiple of the class number h−p of the number field \({\mathbb Q(\sqrt {-p})}\) . In the case of metacyclic groups of odd order pq, with p, q primes, we show that η(0) is a rational multiple of h−p.

On complete stable minimal surfaces in 4-manifolds with positive isotropic curvature

We prove the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to the complex plane in any complete orientable four-dimensional Riemannian manifold with uniformly positive isotropic curvature. We also generalize the same nonexistence result to higher dimensions provided that the ambient manifold has uniformly positive complex sectional curvature. The proof consists of two parts, assuming an eigenvalue condition on the Cauchy-Riemann operator of a holomorphic bundle, we prove (1) a vanishing theorem for these holomorphic bundles on the complex plane; (2) an existence theorem for holomorphic sections with controlled growth by Hormander's weighted L^2-method.

Infinitesimal isometries on tangent sphere bundles over three-dimensional manifolds

In this article, we study the infinitesimal isometries on tangent sphere bundles over orientable three-dimensional Riemannian manifolds. Focusing on the vector fields which do not preserve fibers, we show the existence of lifts to the bundles of orthonormal frames. These lifts enable us to analyze the infinitesimal isometries by the symmetry of principal fiber bundles. We prove that the tangent sphere bundle admits a non-fiber-preserving infinitesimal isometry if and only if the base manifold has the same constant sectional curvatures as the fibers have. As an application, we classify the infinitesimal isometries on tangent sphere bundles for the three dimensional case.

Soliton solutions of the mean curvature flow and minimal hypersurfaces

Let(M,g)(M,g)be an oriented Riemannian manifold of dimension at least33andX∈X(M)\mathbf {X} \in \mathfrak {X}(M)a vector field. We show that the Monge-Ampère differential system (M.A.S.) forX\mathbf {X}-pseudosoliton hypersurfaces on(M,g)(M,g)is equivalent to the minimal hypersurface M.A.S. on(M,g¯)(M,\bar {g})for some Riemannian metricg¯\bar {g}if and only ifX\mathbf {X}is the gradient of a functionuu, in which caseg¯=e−2ug\bar {g}=e^{-2u}g. Counterexamples to this equivalence for surfaces are also given.

Einstein four-manifolds with skew torsion

We develop a notion of Einstein manifolds with skew torsion on compact, orientable Riemannian manifolds of dimension four. We prove an analogue of the Hitchin–Thorpe inequality and study the case of equality. We use the link with self-duality to study the moduli space of 1-instantons on S 4 for a family of metrics defined by Bonneau.

On the Rabinowitz Floer homology of twisted cotangent bundles

Let (M, g) be a closed connected orientable Riemannian manifold of dimension n ≥ 2. Let ω: = ω0 + π*σ denote a twisted symplectic form on T*M, where \({\sigma\in\Omega^{2}(M)}\) is a closed 2-form and ω0 is the canonical symplectic structure \({dp\wedge dq}\) on T*M. Suppose that σ is weakly exact and its pullback to the universal cover \({\widetilde{M}}\) admits a bounded primitive. Let \({H:T^{*}M\rightarrow\mathbb{R}}\) be a Hamiltonian of the form \({(q,p)\mapsto\frac{1}{2}\left|p\right|^{2}+U(q)}\) for \({U\in C^{\infty}(M,\mathbb{R})}\). Let Σk: = H−1(k), and suppose that k > c(g, σ, U), where c(g, σ, U) denotes the Mane critical value. In this paper we compute the Rabinowitz Floer homology of such hypersurfaces. Under the stronger condition that k > c0(g, σ, U), where c0(g, σ, U) denotes the strict Mane critical value, Abbondandolo and Schwarz (J Topol Anal 1:307–405, 2009) recently computed the Rabinowitz Floer homology of such hypersurfaces, by means of a short exact sequence of chain complexes involving the Rabinowitz Floer chain complex and the Morse (co)chain complex associated to the free time action functional. We extend their results to the weaker case k > c(g, σ, U), thus covering cases where σ is not exact. As a consequence, we deduce that the hypersurface Σk is never (stably) displaceable for any k > c(g, σ, U). This removes the hypothesis of negative curvature in Cieliebak et al. (Geom Topol 14:1765–1870, 2010, Theorem 1.3) and thus answers a conjecture of Cieliebak, Frauenfelder and Paternain raised in Cieliebak et al. (2010). Moreover, following Albers and Frauenfelder (2009; J Topol Anal 2:77–98, 2010) we prove that for k > c(g, σ, U), any \({\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}\) has a leaf-wise intersection point in Σk, and that if in addition \({\dim\, H_{*}(\Lambda M;\mathbb{Z}_{2})=\infty}\), dim M ≥ 2, and the metric g is chosen generically, then for a generic \({\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}\) there exist infinitely many such leaf-wise intersection points.

Existence of solutions for a nonlinear elliptic problem on a Riemannian manifold

Let ( M , g ) be a noncompact, connected, orientable smooth N-dimensional Riemannian manifold without boundary. We consider the existence of solutions of problem (P) Δ g u + a u log u = 0 on M , where Δ g stands for the Laplacian in M and a > 0 .

Supercurrent coupling in the Faddeev–Skyrme model

Motivated by the sigma model limit of multicomponent Ginzburg–Landau theory, a version of the Faddeev–Skyrme model is considered in which the scalar field is coupled dynamically to a one-form field called the supercurrent. This coupled model is investigated in the general setting where physical space M is an oriented Riemannian manifold and the target space N is a Kähler manifold, and its properties are compared with the usual, uncoupled Faddeev–Skyrme model. In the case N = S 2 , it is shown that supercurrent coupling destroys the familiar topological lower energy bound of Vakulenko and Kapitanski when M = R 3 , and the less familiar linear bound when M is a compact 3-manifold. Nonetheless, local energy minimizers may still exist. The first variation formula is derived and used to construct three families of static solutions of the model, all on compact domains. In particular, a coupled version of the unit charge hopfion on a three-sphere of arbitrary radius is found. The second variation formula is derived, and used to analyze the stability of some of these solutions. In particular, it is shown that, in contrast to the uncoupled model, the coupled unit hopfion on the three-sphere of radius R is unstable for all R . This gives an explicit, exact example of supercurrent coupling destabilizing a stable solution of the usual Faddeev–Skyrme model, and casts doubt on the conjecture of Babaev, Faddeev and Niemi that knot solitons should exist in the low energy regime of two-component superconductors.

Hermitian Yang–Mills equations and pseudo-holomorphic bundles on nearly Kähler and nearly Calabi–Yau twistor 6-manifolds

We consider the Hermitian Yang–Mills (HYM) equations for gauge potentials on a complex vector bundle E over an almost complex manifold X 6 which is the twistor space of an oriented Riemannian manifold M 4 . Each solution of the HYM equations on such X 6 defines a pseudo-holomorphic structure on the bundle E . It is shown that the pull-back to X 6 of any anti-self-dual gauge field on M 4 is a solution of the HYM equations on X 6 . This correspondence allows us to introduce new twistor actions for bosonic and supersymmetric Yang–Mills theories. As examples of X 6 we consider homogeneous nearly Kähler and nearly Calabi–Yau manifolds which are twistor spaces of S 4 , C P 2 and B 4 , C B 2 (real 4-ball and complex 2-ball), respectively. Various explicit examples of solutions to the HYM equations on these spaces are provided. Applications in flux compactifications of heterotic strings are briefly discussed.

On the existence of almost contact structure and the contact magnetic field

We give a simple proof of the existence of an almost contact metric structure on any orientable 3-dimensional Riemannian manifold (M3, g) with the prescribed metric g as the adapted metric of the almost contact metric structure. By using the key formula for the structure tensor obtained in the proof this theorem, we give an application which allows us to completely determine the magnetic flow of the contact magnetic field in any 3-dimensional Sasakian manifold.

A first eigenvalue estimate for embedded hypersurfaces

Suppose that M is a compact orientable hypersurface embedded in a compact n-dimensional orientable Riemannian manifold N. Suppose that the Ricci curvature of N is bounded below by a positive constant k. We show that 2 λ 1 > k − ( n − 1 ) max M | H | where λ 1 is the first eigenvalue of the Laplacian of M and H is the mean curvature of M.

Free abelian covers, short loops, stable length, and systolic inequalities

We explore the geometry of the Abel–Jacobi map f from a closed, orientable Riemannian manifold X to its Jacobi torus $$H_1(X;{\mathbb{R}})/ H_1(X;{\mathbb{Z}})_{{\mathbb{R}}}$$ . Applying M. Gromov’s filling inequality to the typical fiber of f, we prove an interpolating inequality for two flavors of shortest length invariants of loops. The procedure works, provided the lift of the fiber is non-trivial in the homology of the maximal free abelian cover, $$\overline{X}$$ , classified by f. We show that the finite-dimensionality of the rational homology of $$\overline{X}$$ is a sufficient condition for the homological non-triviality of the fiber. When applied to nilmanifolds, our “fiberwise” inequality typically gives stronger information than the filling inequality for X itself. In dimension 3, we present a sufficient non-vanishing condition in terms of Massey products. This condition holds for certain manifolds that do not fiber over their Jacobi torus, such as 0-framed surgeries on suitable links. Our systolic inequality applies to surface bundles over the circle (provided the algebraic monodromy has 1-dimensional coinvariants), even though the Massey product invariant vanishes for some of these bundles.

The spectrum of twisted Dirac operators on compact flat manifolds

Let M M be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of M M , and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group Z 2 k \mathbb {Z}_2^k , we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the η \eta -series, in terms of values of Hurwitz zeta functions, and the η \eta -invariant. We give the dimension of the space of harmonic spinors and characterize all Z 2 k \mathbb {Z}_2^k -manifolds having asymmetric Dirac spectrum. Furthermore, we exhibit many examples of Dirac isospectral pairs of Z 2 k \mathbb {Z}_2^k -manifolds which do not satisfy other types of isospectrality. In one of the main examples, we construct a large family of Dirac isospectral compact flat n n -manifolds, pairwise nonhomeomorphic to each other of the order of a n a^n .

Yang–Mills theory for bundle gerbes

Given a bundle gerbe with connection on an oriented Riemannian manifold of dimension at least equal to 3, we formulate and study the associated Yang?Mills equations. When the Riemannian manifold is compact and oriented, we prove the existence of instanton solutions to the equations and also determine the moduli space of instantons, thus giving a complete analysis in this case. We also discuss duality in this context.

Canonical multi-symplectic structure on the total exterior algebra bundle

The aim of this paper is to construct multi-symplectic structures starting with the geometry of an oriented Riemannian manifold, independent of a Lagrangian or a particular partial differential equation (PDE). The principal observation is that on an n -dimensional orientable manifold M there is a canonical quadratic form Θ associated with the total exterior algebra bundle on M . On the fibre, which has dimension 2 n , the form Θ can be locally decomposed into n classical symplectic structures. When concatenated, these n -symplectic structures define a partial differential operator, J ∂ , which turns out to be a Dirac operator with multi-symplectic structure. The operator J ∂ generalizes the product operator J (d/d t ) in classical symplectic geometry, and M is a generalization of the base manifold (i.e. time) in classical Hamiltonian dynamics. The structure generated by Θ provides a natural setting for analysing a class of covariant nonlinear gradient elliptic operators. The operator J ∂ is elliptic, and the generalization of Hamiltonian systems, J ∂ Z =∇ S ( Z ), for a section Z of the total exterior algebra bundle, is also an elliptic PDE. The inverse problem—find S ( Z ) for a given elliptic PDE—is shown to be related to a variant of the Legendre transform on k -forms. The theory is developed for flat base manifolds, but the constructions are coordinate free and generalize to Riemannian manifolds with non-trivial curvature. Some applications and implications of the theory are also discussed.