Real Analytic Riemannian Manifold Research Articles

Algebras of pseudo-differential operators acting on holomorphic Sobolev spaces

We search for pseudo-differential operators acting on holomorphic Sobolev spaces. The operators should mirror the standard Sobolev mapping property in the holomorphic analogues. The setting is a closed real-analytic Riemannian manifold, or Lie group with a bi-invariant metric, and the holomorphic Sobolev spaces are defined on a fixed Grauert tube about the core manifold. We find that every pseudo-differential operator in the commutant of the Laplacian is of this kind. Moreover, so are all the operators in the commutant of certain analytic pseudo-differential operators, but for more general tubes, provided that an old statement of Boutet de Monvel holds true generally. In the Lie group setting, we find even larger algebras, and characterize all their elliptic elements. These latter algebras are determined by global matrix-valued symbols.

The Calderón problem for the conformal Laplacian

We consider a conformally invariant version of the Calderon problem, where the objective is to determine the conformal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map for the conformal Laplacian. The main result states that a locally conformally real-analytic manifold in dimensions $\geq 3$ can be determined in this way, giving a positive answer to an earlier conjecture by Lassas and Uhlmann (2001). The proof proceeds as in the standard Calderon problem on a real-analytic Riemannian manifold, but new features appear due to the conformal structure. In particular, we introduce a new coordinate system that replaces harmonic coordinates when determining the conformal class in a neighborhood of the boundary.

Nodal intersections and geometric control

We prove that the number of nodal points on an $\mathcal{S}$-good real analytic curve $\mathcal{C}$ of a sequence $\mathcal{S}$ of Laplace eigenfunctions $\varphi_j$ of eigenvalue $-\lambda^2_j$ of a real analytic Riemannian manifold $(M, g)$ is bounded above by $A_{g , \mathcal{C}} \lambda_j$. Moreover, we prove that the codimension-two Hausdorff measure $\mathcal{H}^{m-2} (\mathcal{N}_{\varphi \lambda} \cap H)$ of nodal intersections with a connected, irreducible real analytic hypersurface $H \subset M$ is $\leq A_{g, H} \lambda_j$. The $\mathcal{S}$-goodness condition is that the sequence of normalized logarithms $\frac{1}{\lambda_j} \operatorname{log} {\lvert \varphi_j \rvert}^2$ does not tend to $-\infty$ uniformly on $\mathcal{C}$, resp. $H$. We further show that a hypersurface satisfying a geometric control condition is $\mathcal{S}$-good for a density one subsequence of eigenfunctions. This gives a partial answer to a question of Bourgain–Rudnick about hypersurfaces on which a sequence of eigenfunctions can vanish. The partial answer characterizes hypersurfaces on which a positive density sequence can vanish or just have $L^2$ norms tending to zero.

Strongly homotopy Lie algebras and deformations of calibrated submanifolds

For an element $\Psi$ in the graded vector space $\Omega^*(M, TM)$ of tangent bundle valued forms on a smooth manifold $M$, a $\Psi$-submanifold is defined as a submanifold $N$ of $M$ such that $\Psi_{|N} \in \Omega^*(N, TN)$. The class of $\Psi$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compact Lie groups. The graded vector space $\Omega^*(M, TM)$ carries a natural graded Lie algebra structure, given by the Frolicher-Nijenhuis bracket $[-,- ]^{FN}$. When $\Psi$ is an odd degree element with $[ \Psi, \Psi]^{FN} =0$, we associate to a $\Psi$-submanifold $N$ a strongly homotopy Lie algebra, which governs the formal and (under certain assumptions) smooth deformations of $N$ as a $\Psi$-submanifold, and we show that under certain assumptions these deformations form an analytic variety. As an application we revisit formal and smooth deformation theory of complex closed submanifolds and of $\varphi$-calibrated closed submanifolds, where $\varphi$ is a parallel form in a real analytic Riemannian manifold.

On Riemannian submersions

We prove that the image of a real analytic Riemannian manifold under a smooth Riemannian submersion is necessarily real analytic.

The Poisson embedding approach to the Calder\xf3n problem

We introduce a new approach to the anisotropic Calderón problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calderón type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result of Lassas et al. (Annales de l’ ENS 34(5):771–787, 2001) solving the Calderón problem on real analytic Riemannian manifolds. The proof uses the Poisson embedding to determine the harmonic functions in the manifold up to a harmonic morphism. The method also involves various Runge approximation results for linear elliptic equations.

Log-scale equidistribution of nodal sets in Grauert tubes

Let Mτ0 be the Grauert tube (of some fixed radius τ0) of a compact, negatively curved, real analytic Riemannian manifold M without boundary. Let φλ be a Laplacian eigenfunction on M of eigenvalue −λ2 and let φλC be its holomorphic extension to Mτ0. In this article, we prove that on Mτ0∖M, there exists a dimensional constant α>0 and a full density subsequence {λjk}k=1∞ of the spectrum for which the masses of the complexified eigenfunctions φλjkC are asymptotically equidistributed at length scale (log⁡λjk)−α. Moreover, the complex zeros of φλjkC also become equidistributed on this logarithmic length scale.

Relative energy gap for harmonic maps of Riemann surfaces into real analytic Riemannian manifolds

We extend the well-known Sacks–Uhlenbeck energy gap result for harmonic maps from closed Riemann surfaces into closed Riemannian manifolds from the case of maps with small energy (thus near a constant map), to the case of harmonic maps with high absolute energy but small energy relative to a reference harmonic map.

Left-invariant grauert tubes on SU(2)

Let M be a real analytic Riemannian manifold. An adapted complex structure on TM is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of TM. We call such manifolds Grauert tubes, or simply tubes. We consider here the case of M = G a compact connected Lie group with a left-invariant metric, and try to determine for which such metrics the associated tube is entire. It is well-known that the Grauert tube of a bi-invariant metric on a Lie group is entire. The case of the smallest group SU(2) is treated completely, thanks to the complete integrability of the geodesic flow for such a metric, a standard result in classical mechanics. Along the way we find a new obstruction to tubes being entire which is made visible by the complete integrability. (New reference and exposition shortened, 11/17/2017.)

Pointwise Bounds for Steklov Eigenfunctions

Let $$(\Omega ,g)$$ be a compact, real-analytic Riemannian manifold with real-analytic boundary $$\partial \Omega .$$ The harmonic extensions of the boundary Dirichlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfunctions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary. The proof uses the Poisson representation for the Steklov eigenfunctions combined with sharp h-microlocal concentration estimates for the boundary Dirichlet-to-Neumann eigenfunctions near the cosphere bundle $$S^*\partial \Omega .$$ These estimates follow from sharp estimates on the concentration of the FBI transforms of solutions to analytic pseudodifferential equations $$Pu=0$$ near the characteristic set $$\{\sigma (P)=0\}$$ .

Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary

ABSTRACTWe prove that the Cauchy data of Dirichlet or Neumann Δ- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e., a point at which a positive measure of geodesics leaving the point return to the point. In the case of real analytic Riemannian manifolds with real analytic boundary, maximal sup norm bounds on boundary traces of eigenfunctions can only be achieved if there exists a point on the boundary at which all geodesics loop back. As an application, the Dirichlet or Neumann eigenfunctions of Riemannian manifolds with concave boundary and non-positive curvature never have eigenfunctions whose boundary traces achieve maximal sup norm bounds. The key new ingredient is the Melrose–Taylor diffractive parametrix and Melrose’s analysis of the Weyl law.

The complex Monge–Ampère equation, Zoll metrics and algebraization

Let M be a real analytic Riemannian manifold. An adapted complex structure on TM is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of TM. We prove here that the only real analytic Zoll metric on the n-sphere with an entire adapted complex structure on TM is the round sphere. Using similar ideas, we answer a special case of an algebraization question raised by the first author, characterizing some Stein manifolds as affine algebraic in terms of plurisubharmonic exhaustion functions satisfying the homogeneous complex Monge–Ampere equation. The result presented here is an extension to higher dimensions of an observation attributed to W. Stoll for the case of Riemann surfaces.

Focal points and sup-norms of eigenfunctions

If (M,g) is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order o(λ) saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is that there exists a self-focal point x0∈M for the geodesic flow at which the associated Perron–Frobenius operator Ux0:L2(S∗x0M)→L2(S∗x0M) has a nontrivial invariant L2 function. The proof is based on an explicit Duistermaat–Guillemin–Safarov pre-trace formula and von Neumann's ergodic theorem.

Cauchy problems for Lorentzian manifolds with special holonomy

On a Lorentzian manifold the existence of a parallel null vector field implies certain constraint conditions on the induced Riemannian geometry of a space-like hypersurface. We will derive these constraint conditions and, conversely, show that every real analytic Riemannian manifold satisfying the constraint conditions can be extended to a Lorentzian manifold with a parallel null vector. Similarly, every parallel null spinor on a Lorentzian manifold induces a generalised imaginary Killing spinor on a space-like hypersurface. Then, using the fact that a parallel spinor field induces a parallel vector field, we can apply the first result to prove: every real analytic Riemannian manifold carrying a real analytic, imaginary generalised Killing spinor can be extended to a Lorentzian manifold with a parallel null spinor. Finally, we give examples of geodesically complete Riemannian manifolds satisfying the constraint conditions.

The Poisson transform on a compact real analytic Riemannian manifold

We study the transform mapping an \(L^{2}\) function f on a compact, real analytic Riemannian manifold X to the analytic continuation of \(\exp (-t \sqrt{\Delta })f\) to the interior of a Grauert tube tube \(M_{t}\) about X. We show that after precomposing with an elliptic pseudodifferential operator this becomes a unitary map from \(L^{2}(X)\) onto the holomorphic \(L^{2}\) functions on \(M_{t}\). If a compact Lie group of isometries acts transitively on X then the inverse of this unitarized map can be constructed by the restrict ion principle.

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.

A new look at adapted complex structures

Given a closed real analytic Riemannian manifold, we construct and study a one parameter family of adapted complex structures on the manifold of its geodesics.

Adapted complex structures and the geodesic flow

In this paper, we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact, real-analytic Riemannian manifold. Motivated by the “complexifier” approach of T. Thiemann as well as certain formulas of V. Guillemin and M. Stenzel, we obtain the polarization associated to the adapted complex structure by applying the “imaginary-time geodesic flow” to the vertical polarization. Meanwhile, at the level of functions, we show that every holomorphic function is obtained from a function that is constant along the fibers by “composition with the imaginary-time geodesic flow.” We give several equivalent interpretations of this composition, including a convergent power series in the vector field generating the geodesic flow.

THE VOLUME OF A LOCAL NODAL DOMAIN

Let M either be a closed real analytic Riemannian manifold or a closed C∞-Riemannian surface. We estimate from below the volume of a nodal domain component in an arbitrary ball, provided that this component enters the ball deeply enough.

Differentiability of the isoperimetric profile and topology of analytic Riemannian manifolds

We show that smooth isoperimetric profiles are exceptional for real analytic Riemannian manifolds. For instance, under some extra assumptions, this can happen only on topological spheres. To cite this article: R. Grimaldi et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).