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 consider the problem of minimising the -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 and any sequence of minimisers converges to the ball of unit perimeter or diameter respectively as . 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 . We also consider these problems for Robin eigenvalues and mixed Dirichlet-Neumann eigenvalues, under an additional geometric constraint.

Let be a smooth domain. We establish conditions under which a weakly conformal, branched -free boundary Hamiltonian stationary Lagrangian immersion of a disc in is a -free boundary minimal immersion. We deduce that if is a weakly conformal, branched -free boundary Hamiltonian stationary Lagrangian immersion of a disc with Legendrian boundary, then is a Lagrangian equatorial plane disc. Furthermore, we present examples of -free boundary Hamiltonian stationary discs, demonstrating the optimality of our assumptions.

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 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 ; the minimal case ( ) of this result was obtained in Chodosh et al. (Ars Inveniendi Analytica, 2022) .

We apply the direct method of the calculus of variations to show that any nonplanar Frenet curve in can be extended to an infinitely narrow flat ribbon having bending energy. We also show that, in general, minimizers are not free of planar points, yet such points must be isolated under the mild condition that the torsion does not vanish.

We establish Pólya-Szegő-type inequalities (PSIs) for Sobolev-functions defined on a regular -dimensional submanifold (possibly with boundary) of a -dimensional Euclidean space, under explicit upper bounds of the total mean curvature. The -Sobolev and Gagliardo-Nirenberg inequalities, as well as the spectral gap in are derived as corollaries. Using these PSIs, we prove a sharp -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 is provided. A second equivalent version of our PSIs is presented in the appendix of this paper, introducing the notion of model space of dimension and total mean curvature bounded by .

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.

This article aims to understand the behavior of the curvature operator of the second kind under the Ricci flow in dimension three. First, we express the eigenvalues of the curvature operator of the second kind explicitly in terms of that of the curvature operator (of the first kind). Second, we prove that -positive/ -nonnegative curvature operator of the second kind is preserved by the Ricci flow in dimension three for all .

The -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 -Assouad dimensions which help to clarify their behaviour. We prove for any bounded doubling metric space and satisfying that there is a function so that the -Assouad dimension of is equal to . We further show that the "upper" variant of the dimension is fully determined by the -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 -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 -Assouad dimensions of overlapping self-similar sets and decreasing sequences with decreasing gaps.

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

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 article, we study the microlocal properties of the geodesic ray transform of symmetric -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 ( 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 DO and each of the FIO components of the normal operator acting on symmetric -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.

CR functions on an embedded quadric always extend holomorphically to where is the closure of the convex hull of the image of the Levi form. When is a closed polygonal cone, we show that the Bergman kernel on the interior of is a derivative of the Szegö kernel. Moreover, we develop the Hardy space theory which turns out to be particularly robust. We provide examples that show that it is unclear how to formulate a corresponding relationship between the Bergman and Szegö kernels on a wider class of quadrics.

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.

Multi-material design optimization problems can, after discretization, be solved by the iterative solution of simpler sub-problems which approximate the original problem at an expansion point to first order. In particular, models constructed from convex separable first order approximations have a long and successful tradition in the design optimization community and have led to powerful optimization tools like the prominently used method of moving asymptotes (MMA). In this paper, we introduce several new separable approximations to a model problem and examine them in terms of accuracy and fast evaluation. The models can, in general, be nonconvex and are based on the Sherman-Morrison-Woodbury matrix identity on the one hand, and on the mathematical concept of topological derivatives on the other hand. We show a surprising relation between two models originating from these two-at a first sight-very different concepts. Numerical experiments show a high level of accuracy for two of our proposed models while also their evaluation can be performed efficiently once enough data has been precomputed in an offline stage. Additionally it is demonstrated that suboptimal decisions can be avoided using our most accurate models.

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.

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.

In this paper we present the following result on regularity of solutions of the second order parabolic equation on cylindrical domains of the form where is a uniform domain (it satisfies both interior corkscrew and Harnack chain conditions) and has a boundary that is -Ahlfors regular. Let be a solution of such PDE in and the non-tangential maximal function of its gradient in spatial directions belongs to for some . Furthermore, assume that for we have that . Then both and also belong to , where and 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 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.

Suppose that , , is a uniform domain with -Ahlfors regular boundary and is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in . We show that the corresponding elliptic measure is quantitatively absolutely continuous with respect to surface measure of in the sense that if and only if any bounded solution to in is -approximable for any . By -approximability of we mean that there exists a function such that and the measure with is a Carleson measure with control over the Carleson norm. As a consequence of this approximability result, we show that boundary 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 -type Carleson measure estimates with 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.

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 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.