Journal of Geometric Analysis最新文献

We show that the direct product of two Stein manifolds with the Hamiltonian density property enjoys the Hamiltonian density property as well. We investigate the relation between the Hamiltonian density property and the symplectic density property. We then establish the Hamiltonian and the symplectic density property for ${\left(C^{\ast }\right)}^{2n}$ and for the so-called traceless Calogero-Moser spaces. As an application we obtain a Carleman-type approximation for Hamiltonian diffeomorphisms of a real form of the traceless Calogero-Moser space.

Let $\Omega \subset C^2$ be a smooth domain. We establish conditions under which a weakly conformal, branched $\Omega$ -free boundary Hamiltonian stationary Lagrangian immersion u of a disc in $C^2$ is a $\Omega$ -free boundary minimal immersion. We deduce that if $u$ is a weakly conformal, branched $B_1\left(0\right)$ -free boundary Hamiltonian stationary Lagrangian immersion of a disc with Legendrian boundary, then $u\left(D^2\right)$ is a Lagrangian equatorial plane disc. Furthermore, we present examples of $\Omega$ -free boundary Hamiltonian stationary discs, demonstrating the optimality of our assumptions.

We obtain geometric lower bounds for the low Steklov eigenvalues of finite-volume hyperbolic surfaces with geodesic boundary. The bounds we obtain depend on the length of a shortest multi-geodesic disconnecting the surfaces into connected components each containing a boundary component and the rate of dependency on it is sharp. Our result also identifies situations when the bound is independent of the length of this multi-geodesic. The bounds also hold when the Gaussian curvature is bounded between two negative constants and can be viewed as a counterpart of the well-known Schoen-Wolpert-Yau inequality for Laplace eigenvalues. The proof is based on analysing the behaviour of the corresponding Steklov eigenfunction on an adapted version of thick-thin decomposition for hyperbolic surfaces with geodesic boundary. Our results extend and improve the previously known result in the compact case obtained by a different method.

We generalize the Cheeger inequality, a lower bound on the first nontrivial eigenvalue of a Laplacian, to the case of geometric sub-Laplacians on rank-varying Carnot-Carathéodory spaces and we describe a concrete method to lower bound the Cheeger constant. The proof is geometric, and works for Dirichlet, Neumann and mixed boundary conditions. One of the main technical tools in the proof is a generalization of Courant's nodal domain theorem, which is proven from scratch for Neumann and mixed boundary conditions. Carnot groups and the Baouendi-Grushin cylinder are treated as examples.

We prove that the reconstruction of a certain type of length spaces from their travel time data on a closed subset is Lipschitz stable. The travel time data is the set of distance functions from the entire space, measured on the chosen closed subset. The case of a Riemannian manifold with boundary with the boundary as the measurement set appears is a classical geometric inverse problem arising from Gel'fand's inverse boundary spectral problem. Examples of spaces satisfying our assumptions include some non-simple Riemannian manifolds, Euclidean domains with non-trivial topology, and metric trees.

We consider the problem of minimising the k-th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are well-posed for Dirichlet eigenvalues in any dimension $d\ge 2$ and any sequence of minimisers converges to the ball of unit perimeter or diameter respectively as $k\to +\infty$ . In this paper, we show that the same is true in the case of Neumann eigenvalues under diameter constraint in any dimension and under perimeter constraint in dimension $d=2$ . We also consider these problems for Robin eigenvalues and mixed Dirichlet-Neumann eigenvalues, under an additional geometric constraint.

The $\varphi$ -Assouad dimensions are a family of dimensions which interpolate between the upper box and Assouad dimensions. They are a generalization of the well-studied Assouad spectrum with a more general form of scale sensitivity that is often closely related to "phase-transition" phenomena in sets. In this article we establish a number of key properties of the $\varphi$ -Assouad dimensions which help to clarify their behaviour. We prove for any bounded doubling metric space F and $\alpha \in R$ satisfying ${\overline{\text{dim}}}_{\text{B}}F<\alpha \le {\text{dim}}_{\text{A}}F$ that there is a function $\varphi$ so that the $\varphi$ -Assouad dimension of F is equal to $\alpha$ . We further show that the "upper" variant of the dimension is fully determined by the $\varphi$ -Assouad dimension, and that homogeneous Moran sets are in a certain sense generic for these dimensions. Further, we study explicit examples of sets where the Assouad spectrum does not reach the Assouad dimension. We prove a precise formula for the $\varphi$ -Assouad dimensions for the boundary of Galton-Watson trees that correspond to a general class of stochastically self-similar sets, including Mandelbrot percolation. The proof of this result combines a sharp large deviations theorem for Galton-Watson processes with bounded offspring distribution and a general Borel-Cantelli-type lemma for infinite structures in random trees. Finally, we obtain results on the $\varphi$ -Assouad dimensions of overlapping self-similar sets and decreasing sequences with decreasing gaps.

We prove a symbolic calculus for a class of pseudodifferential operators, and discuss its applications to $L^2$ -compactness via a compact version of the T(1) theorem.

In this article, we study the microlocal properties of the geodesic ray transform of symmetric m-tensor fields on 2-dimensional Riemannian manifolds with boundary allowing the possibility of conjugate points. As is known from an earlier work on the geodesic ray transform of functions in the presence of conjugate points, the normal operator can be decomposed into a sum of a pseudodifferential operator ( $\Psi$ DO) and a finite number of Fourier integral operators (FIOs) under the assumption of no singular conjugate pairs along geodesics, which always holds in 2-dimensions. In this work, we use the method of stationary phase to explicitly compute the principal symbol of the $\Psi$ DO and each of the FIO components of the normal operator acting on symmetric m-tensor fields. Next, we construct a parametrix recovering the solenoidal component of the tensor fields modulo FIOs, and prove a cancellation of singularities result, similar to an earlier result of Monard, Stefanov and Uhlmann for the case of geodesic ray transform of functions in 2-dimensions. We point out that this type of cancellation result is only possible in the 2-dimensional case.

In 2018, Marques and Neves proposed a volume preserving intrinsic flat stability conjecture concerning their width rigidity theorem for the unit round 3-sphere. In this work, we establish the validity of this conjecture under the additional assumption of rotational symmetry. Furthermore, we obtain a rigidity theorem in dimensions at least three for rotationally symmetric manifolds, which is analogous to the width rigidity theorem of Marques and Neves. We also prove a volume preserving intrinsic flat stability result for this rigidity theorem. Lastly, we study variants of Marques and Neves' stability conjecture. In the first, we show Gromov-Hausdorff convergence outside of certain "bad" sets. In the second, we assume non-negative Ricci curvature and show Gromov-Hausdorff stability.