Journal of Geometric Analysis最新文献

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.

In this work we find a unifying scheme for the known explicit complex-valued eigenfunctions on the classical compact Riemannian symmetric spaces. For this we employ the well-known Cartan embedding for those spaces. This also leads to the construction of new eigenfunctions on the quaternionic Grassmannians.

We show a local rigidity result for the integrability of symplectic billiards. We prove that any domain which is close to an ellipse, and for which the symplectic billiard map is rationally integrable must be an ellipse as well. This is in spirit of the result of [2] for Birkhoff billiards.

In this note we prove the existence of two proper biharmonic maps between the Euclidean ball of dimension bigger than four and Euclidean spheres of appropriate dimensions. We will also show that, in low dimensions, both maps are unstable critical points of the bienergy.

We establish Pólya-Szegő-type inequalities (PSIs) for Sobolev-functions defined on a regular n-dimensional submanifold $\Sigma$ (possibly with boundary) of a $(n+m)$ -dimensional Euclidean space, under explicit upper bounds of the total mean curvature. The p-Sobolev and Gagliardo-Nirenberg inequalities, as well as the spectral gap in $W_0^{1,p}\left(\Sigma \right)$ are derived as corollaries. Using these PSIs, we prove a sharp p-Log-Sobolev inequality for minimal submanifolds in codimension one and two. The asymptotic sharpness of both the multiplicative constant appearing in PSIs and the assumption on the total mean curvature bound as $n\to \infty$ is provided. A second equivalent version of our PSIs is presented in the appendix of this paper, introducing the notion of model space $(R^{+},m_{n,K})$ of dimension n and total mean curvature bounded by K.

In compact Riemannian manifolds of dimension 3 or higher with positive Ricci curvature, we prove that every constant mean curvature hypersurface produced by the Allen-Cahn min-max procedure in Bellettini and Wickramasekera (arXiv:2010.05847, 2020) (with constant prescribing function) is a local minimiser of the natural area-type functional around each isolated singularity. In particular, every tangent cone at each isolated singularity of the resulting hypersurface is area-minimising. As a consequence, for any real $\lambda$ we show, through a surgery procedure, that for a generic 8-dimensional compact Riemannian manifold with positive Ricci curvature there exists a closed embedded smooth hypersurface of constant mean curvature $\lambda$ ; the minimal case ( $\lambda =0$ ) of this result was obtained in Chodosh et al. (Ars Inveniendi Analytica, 2022) .

In this paper we present the following result on regularity of solutions of the second order parabolic equation ${\partial }_{t}u-\text{div}\left(A\nabla u\right)+B·\nabla u=0$ on cylindrical domains of the form $\Omega =O\times R$ where $O\subset R^n$ is a uniform domain (it satisfies both interior corkscrew and Harnack chain conditions) and has a boundary that is $n-1$ -Ahlfors regular. Let u be a solution of such PDE in $\Omega$ and the non-tangential maximal function of its gradient in spatial directions $\tilde{N}\left(\nabla u\right)$ belongs to $L^p\left(\partial \Omega \right)$ for some $p>1$ . Furthermore, assume that for ${u|}_{\partial \Omega }=f$ we have that $D_t^{1/2}f\in L^p\left(\partial \Omega \right)$ . Then both $\tilde{N}\left(D_t^{1/2}u\right)$ and $\tilde{N}\left(D_t^{1/2}{H}_{t}u\right)$ also belong to $L^p\left(\partial \Omega \right)$ , where $D_t^{1/2}$ and $H_t$ are the half-derivative and the Hilbert transform in the time variable, respectively. We expect this result will spur new developments in the study of solvability of the $L^p$ parabolic Regularity problem as thanks to it it is now possible to formulate the parabolic Regularity problem on a large class of time-varying domains.

We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional-order $q\in [0,\infty )$. We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if $q>1/2$. Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if $q>3/2$, whereas if $q<3/2$ then finite-time blowup may occur. The geodesic completeness for $q>3/2$ is obtained by proving metric completeness of the space of $H^q$-immersed curves with the distance induced by the Riemannian metric.

Suppose that $\Omega \subset R^{n+1}$, $n\ge 1$, is a uniform domain with n-Ahlfors regular boundary and L is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in $\Omega$. We show that the corresponding elliptic measure ${\omega }_L$ is quantitatively absolutely continuous with respect to surface measure of $\partial \Omega$ in the sense that ${\omega }_L\in A_{\infty }\left(\sigma \right)$ if and only if any bounded solution u to $Lu=0$ in $\Omega$ is $\epsilon$-approximable for any $\epsilon \in (0,1)$. By $\epsilon$-approximability of u we mean that there exists a function $\Phi ={\Phi }^{\epsilon }$ such that ${\Vert u-\Phi \Vert }_{L^{\infty }\left(\Omega \right)}\le \epsilon {\Vert u\Vert }_{L^{\infty }\left(\Omega \right)}$ and the measure ${\tilde{\mu }}_{\Phi }$ with $d\tilde{\mu }=\left|\nabla \Phi \left(Y\right)\right|dY$ is a Carleson measure with $L^{\infty }$ control over the Carleson norm. As a consequence of this approximability result, we show that boundary $\text{BMO}$ functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy $L^1$-type Carleson measure estimates with $\text{BMO}$ control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis.