Submanifolds Of Codimension Research Articles

Non‐degenerate minimal submanifolds as energy concentration sets: A variational approach

AbstractWe prove that every non‐degenerate minimal submanifold of codimension two can be obtained as the energy concentration set of a family of critical maps for the (rescaled) Ginzburg–Landau functional. The proof is purely variational, and follows the strategy laid out by Jerrard and Sternberg, extending a recent result for geodesics by Colinet–Jerrard–Sternberg. The same proof applies also to the ‐Yang–Mills–Higgs and to the Allen–Cahn–Hilliard energies. While for the latter energies gluing methods are also effective, in general dimension our proof is by now the only available one in the Ginzburg–Landau setting, where the weaker energy concentration is the main technical difficulty.

Rigorous derivation of discrete fracture models for Darcy flow in the limit of vanishing aperture

<abstract><p>We consider single-phase flow in a fractured porous medium governed by Darcy's law with spatially varying hydraulic conductivity matrices in both bulk and fractures. The width-to-length ratio of a fracture is of the order of a small parameter $ \varepsilon $ and the ratio $ {{K_\mathrm{f}}}^\star / {{K_\mathrm{b}}}^\star $ of the characteristic hydraulic conductivities in the fracture and bulk domains is assumed to scale with $ \varepsilon^\alpha $ for a parameter $ \alpha \in \mathbb{R} $. The fracture geometry is parameterized by aperture functions on a submanifold of codimension one. Given a fracture, we derive the limit models as $ \varepsilon \rightarrow 0 $. Depending on the value of $ \alpha $, we obtain five different limit models as $ \varepsilon \rightarrow 0 $, for which we present rigorous convergence results.</p></abstract>

On Tangent Bundles of Submanifolds of a Riemannian Manifold Endowed with a Quarter-Symmetric Non-metric Connection

The object of this article is to study a quarter-symmetric non-metric connection in the tangent bundle and induced metric and connection on submanifold of co-dimension 2 and hypersurface concerning the quarter-symmetric non-metric connection in the tangent bundle. The Weingarten equations concerning the quarter-symmetric non-metric connection on a submanifold of co-dimension 2 and the hypersurface in the tangent bundle are obtained. Finally, authors deduce the Riemannian curvature tensor and Gauss and Codazzi equations on a submanifold of co-dimension 2 and hypersurface of the Riemannian manifold concerning the quarter-symmetric non-metric connection in the tangent bundle.

Convergence of the self‐dual U(1)‐Yang–Mills–Higgs energies to the (n−2)$(n-2)$‐area functional

AbstractGiven a hermitian line bundle on a closed Riemannian manifold , the self‐dual Yang–Mills–Higgs energies are a natural family of functionals defined for couples consisting of a section and a hermitian connection ∇ with curvature . While the critical points of these functionals have been well‐studied in dimension two by the gauge theory community, it was shown in [52] that critical points in higher dimension converge as (in an appropriate sense) to minimal submanifolds of codimension two, with strong parallels to the correspondence between the Allen–Cahn equations and minimal hypersurfaces. In this paper, we complement this idea by showing the Γ‐convergence of to (2π times) the codimension two area: more precisely, given a family of couples with , we prove that a suitable gauge invariant Jacobian converges to an integral ‐cycle Γ, in the homology class dual to the Euler class , with mass . We also obtain a recovery sequence, for any integral cycle in this homology class. Finally, we apply these techniques to compare min‐max values for the ‐area from the Almgren–Pitts theory with those obtained from the Yang–Mills–Higgs framework, showing that the former values always provide a lower bound for the latter. As an ingredient, we also establish a Huisken‐type monotonicity result along the gradient flow of .

Proof of the Michael–Simon–Sobolev inequality using optimal transport

Abstract We give an alternative proof of the Michael–Simon–Sobolev inequality using techniques from optimal transport. The inequality is sharp for submanifolds of codimension 2.

Nonnegative scalar curvature on manifolds with at least two ends

AbstractLet be an orientable connected ‐dimensional manifold with and let be a two‐sided closed connected incompressible hypersurface that does not admit a metric of positive scalar curvature (abbreviated by psc). Moreover, suppose that the universal covers of and are either both spin or both nonspin. Using Gromov's ‐bubbles, we show that does not admit a complete metric of psc. We provide an example showing that the spin/nonspin hypothesis cannot be dropped from the statement of this result. This answers, up to dimension 7, a question by Gromov for a large class of cases. Furthermore, we prove a related result for submanifolds of codimension 2. We deduce as special cases that, if does not admit a metric of psc and , then does not carry a complete metric of psc and does not carry a complete metric of uniformly psc, provided that and , respectively. This solves, up to dimension 7, a conjecture due to Rosenberg and Stolz in the case of orientable manifolds.

Real Kaehler submanifolds in codimension up to four

Let f\colon M^{2n}\to\mathbb{R}^{2n+4} be an isometric immersion of a Kaehler manifold of complex dimension n\geq 5 into Euclidean space with complex rank at least 5 everywhere. Our main result is that, along each connected component of an open dense subset of M^{2n} , either f is holomorphic in \mathbb{R}^{2n+4}\cong\mathbb{C}^{n+2} , or it is in a unique way a composition f=F\circ h of isometric immersions. In the latter case, we have that h\colon M^{2n}\to N^{2n+2} is holomorphic and F\colon N^{2n+2}\to\mathbb{R}^{2n+4} belongs to the class, by now quite well understood, of non-holomorphic Kaehler submanifolds in codimension two. Moreover, the submanifold F is minimal if and only if f is minimal.

Arf invariants of codimension one in a Wall group of the dihedral group

An element $x$ is specified in the Wall group $L_3(D_3)$ of the dihedral group of order $8$ with trivial orientation character, such that $x$ is an element of the third type in the sense of Kharshiladze with respect to any system of one-sided submanifolds of codimension $1$ for which the splitting obstruction group along the first submanifold is isomorphic to $LN_1(\mathbb Z/2\oplus \mathbb Z/2\to D_3)$. The element $x$ is not realisable as an obstruction to surgery on a closed $\mathrm{PL}$-manifold. It is also proved that the unique nontrivial element of the group $LN_3(\mathbb Z/2\oplus \mathbb Z/2\to D_3^-)$ can be detected using the Hasse-Witt $Wh_2$-torsion. Bibliography: 25 titles.

Geometric Invariants for a Class of Submodules of Analytic Hilbert Modules Via the Sheaf Model

Let \(\Omega \subseteq {\mathbb {C}}^m\) be a bounded connected open set and \({\mathcal {H}} \subseteq {\mathcal {O}}(\Omega )\) be an analytic Hilbert module, i.e., the Hilbert space \({\mathcal {H}}\) possesses a reproducing kernel K, the polynomial ring \(\mathbb C[{\varvec{z}}]\subseteq {\mathcal {H}}\) is dense and the point-wise multiplication induced by \(p\in {\mathbb {C}}[{\varvec{z}}]\) is bounded on \({\mathcal {H}}\). We fix an ideal \({\mathcal {I}} \subseteq {\mathbb {C}}[{\varvec{z}}]\) generated by \(p_1,\ldots ,p_t\) and let \([{\mathcal {I}}]\) denote the completion of \({\mathcal {I}}\) in \(\mathcal H\). The sheaf \({\mathcal {S}}^{\mathcal {H}}\) associated to analytic Hilbert module \({\mathcal {H}}\) is the sheaf \({\mathcal {O}}(\Omega )\) of holomorphic functions on \(\Omega \) and hence is free. However, the subsheaf \({\mathcal {S}}^{\mathcal [{\mathcal {I}}]}\) associated to \([{\mathcal {I}}]\) is coherent and not necessarily locally free. Building on the earlier work of Biswas, Misra and Putinar (Journal fr die reine und angewandte Mathematik (Crelles Journal) 662:165–204, 2012), we prescribe a hermitian structure for a coherent sheaf and use it to find tractable invariants. Moreover, we prove that if the zero set \(V_{[{\mathcal {I}}]}\) is a submanifold of codimension t, then there is a unique local decomposition for the kernel \(K_{[{\mathcal {I}}]}\) along the zero set that serves as a holomorphic frame for a vector bundle on \(V_{[{\mathcal {I}}]}\). The complex geometric invariants of this vector bundle are also unitary invariants for the submodule \([{\mathcal {I}}] \subseteq {\mathcal {H}}\).

Spectral geometry of nuts and bolts

We study the spectrum of Laplace operators on a one-parameter family of gravitational instantons of bi-axial Bianchi IX type coupled to an abelian connection with self-dual curvature. The family of geometries includes the Taub-NUT (TN), Taub-bolt and Euclidean Schwarzschild geometries and interpolates between them. The interpolating geometries have conical singularities along a submanifold of co-dimension two, but we prove that the associated Laplace operators have natural selfadjoint extensions and study their spectra. In particular, we determine the essential spectrum and prove that its complement, the discrete spectrum, is infinite. We compute some of these eigenvalues numerically and compare the numerical results with an analytical approximation derived from the asymptotic TN form of each of the geometries in our family.

A Class of Einstein Submanifolds of Euclidean Space

In this paper we give local and global parametric classifications of a class of Einstein submanifolds of Euclidean space. The highlight is for submanifolds of codimension two since in this case the assumptions are only of intrinsic nature.

On Homotopy Classification of Elliptic Problems with Contractions and K-Groups of Corresponding C*-Algebras

We consider a computation of a group of stable homotopy classes for pseudodifferential elliptic boundary problems. We study this problem in terms of topological K-groups of spaces in the following cases: boundary-value problems on manifolds with boundary, transmission problems with conditions on a closed submanifold of codimension one, and nonlocal problems with contractions.

Construction of counterexamples to the 2–jet determination Chern–Moser Theorem in higher codimension

We first construct a counterexample of a generic quadratic submanifold of codimension $5$ in $\Bbb C^9$ which admits a real analytic infinitesimal CR automorphism with homogeneous polynomial coefficients of degree $4.$ This example also resolves a question in the Tanaka prolongation theory that was open for more than 50 years. Then we give sufficient conditions to generate more counterexamples to the $2-$jet determination Chern-Moser Theorem in higher codimension. In particular, we construct examples of generic quadratic submanifolds with jet determination of arbitrarily high order.

Sharp endpoint estimates for eigenfunctions restricted to submanifolds of codimension 2

Burq-Gérard-Tzvetkov [10] and Hu [26] established Lp estimates (2≤p≤∞) for the restriction of eigenfunctions to submanifolds. The estimates are sharp, except for the log loss at the endpoint L2 estimates for submanifolds of codimension 2. It has long been believed that the log loss at the endpoint can be removed in general, while the problem is still open. So this paper is devoted to the study of sharp endpoint restriction estimates for eigenfunctions in this case. Chen-Sogge [17] removed the log loss for the geodesics on 3-dimensional manifolds. In this paper, we generalize their result to higher dimensions and prove that the log loss can be removed for totally geodesic submanifolds of codimension 2. Moreover, on 3-dimensional manifolds, we can remove the log loss for curves with nonvanishing geodesic curvatures, and more general finite type curves. The problem in 3D is essentially related to Hilbert transforms along curves in the plane and a class of singular oscillatory integrals studied by Phong-Stein [35], Ricci-Stein [38], Pan [32], Seeger [40], Carbery-Pérez [15].

A CR singular analogue of Severi’s theorem

Real-analytic CR functions on real-analytic CR singular submanifolds are not in general restrictions of holomorphic functions, unlike in the CR nonsingular case. We give a simple condition that completely characterizes those quadric CR singular manifolds of codimension 2 in ${\mathbb C}^{n+1}$ for which an extension result holds. Consequently, we obtain an extension result for general real-analytic CR singular submanifolds of codimension 2. As applications we give a condition for the flattening of such submanifolds, and we classify CR singular images of CR submanifolds up to second order.

Photon surfaces in less symmetric spacetimes

We investigate photon surfaces and their stability in a less symmetric spacetime, a general static warped product with a warping function acting on a Riemannian submanifold of codimension two. We find a one-dimensional pseudopotential that gives photon surfaces as its extrema regardless of the spatial symmetry of the submanifold. The maxima and minima correspond to unstable and stable photon surfaces, respectively. It is analogous to the potential giving null circular orbits in a spherically symmetric spacetime. We also see that photon surfaces indeed exist for the spacetimes which are solutions to the Einstein equation. The parameter values for which the photon surfaces exist are specified. As we show finally, the pseudopotential arises due to the separability of the null geodesic equation, and the separability comes from the existence of a Killing tensor in the spacetime. The result leads to the conclusion that photon surfaces may exist even in a less symmetric spacetime if the spacetime admits a Killing tensor.

On special submanifolds of the Page space

In this paper, we study some classes of submanifolds of codimension one and two in the Page space. These submanifolds are totally geodesic. We also compute their curvature and show that some of them are constant curvature spaces. Finally, we give information on how the Page space is related to some other metrics on the same underlying smooth manifold.

On metrics with minimal singularities of line bundles whose stable base loci admit holomorphic tubular neighborhoods

We investigate the minimal singularities of metrics on a big line bundle $L$ over a projective manifold when the stable base locus $Y$ of $L$ is a submanifold of codimension $r\geq 1$. Under some assumptions on the normal bundle and a neighborhood of $Y$, we give a explicit description of the minimal singularity of metrics on $L$. We apply this result to study a higher (co-)dimensional analogue of Zariski's example, in which the line bundle $L$ is not semi-ample, however it is nef and big.

On cylindricity of submanifolds of nonnegative Ricci curvature in a Minkowski space

We consider Finsler submanifolds of nonnegative Ricci curvature in a Minkowski space which contain a line or whose relative nullity index is positive. For hypersurfaces, submanifolds of codimension two or of dimension two, we prove that the submanifold is a cylinder, under a certain condition on the inertia of the pencil of the second fundamental forms. We give an example of a surface of positive flag curvature in a three-dimensional Minkowski space which is not locally convex.

Period integrals in nonpositively curved manifolds

We provide an improvement of a half power of log to standard bounds on integrals of Laplace eigenfunctions over submanifolds of codimension 2 or more, where the ambient space is a compact Riemannian manifold with negative sectional curvature. We provide the same improvement for hypersurfaces whose second fundamental form differs sufficiently from that of spheres of infinite radius. This result extends previous ones obtained in the 2-dimensional setting by Chen and Sogge; Sogge, Xi, and Zhang; and the author.