We discuss the essential spectrum of essentially self-adjoint elliptic differential operators of first order and of Laplace type operators on Riemannian vector bundles over geometrically finite orbifolds.

This paper concerns the construction of minimal varieties with small canonical volumes. The first part devotes to establishing an effective nefness criterion for the canonical divisor of a weighted blow-up over a weighted hypersurface, from which we construct plenty of new minimal $3$-folds including 59 families of minimal $3$-folds of general type, several infinite series of minimal $3$-folds of Kodaira dimension $2$, 2 families of minimal $3$-folds of general type on the Noether line, and 12 families of minimal $3$-folds of general type near the Noether line. In the second part, we prove effective lower bounds of canonical volumes of minimal $n$-folds of general type with canonical dimension $n-1$ or $n-2$. Examples are provided to show that the theoretical lower bounds are optimal in dimension at most $5$ and nearly optimal in higher dimensions.

We address the classification of ancient solutions to the Gauss curvature flow under the assumption that the solutions are contained in a cylinder of bounded cross-section. For each cylinder of convex bounded cross-section, we show that there are only two ancient solutions which are asymptotic to this cylinder: the non-compact translating soliton and the compact oval solution obtained by gluing two translating solitons approaching each other from time $-\infty$ from two opposite ends.

We apply the method of Lyapunov–Schmidt reduction to study area-constrained Willmore surfaces in Riemannian $3$-manifolds asymptotic to Schwarzschild. In particular, we prove that the end of such a manifold is foliated by distinguished area-constrained Willmore spheres. The leaves are the unique area-constrained Willmore spheres with large area, non-negative Hawking mass, and distance to the center of the manifold at least a small multiple of the area radius. Unlike previous related work, we only require that the scalar curvature satisfies mild asymptotic conditions. We also give explicit examples to show that these conditions on the scalar curvature are necessary.

We explore connections furnished by the Funk metric, a relative of the Hilbert metric, between projective geometry, billiards, convex geometry and affine inequalities. We first show that many metric invariants of the Funk metric are invariant under projective maps, as well as under projective duality. These include the Holmes–Thompson volume and surface area of convex subsets, and the length spectrum of their boundary, extending results of Holmes–Thompson and Álvarez Paiva on Schäffer’s dual girth conjecture. We explore in particular Funk billiards, which generalize hyperbolic billiards in the same way that Minkowski billiards generalize Euclidean ones, and extend a result of Gutkin–Tabachnikov on the duality of Minkowski billiards.

We next consider the volume of outward balls in Funk geometry. We conjecture a general affine inequality corresponding to the volume maximizers, which includes the Blaschke–Santaló and centro-affine isoperimetric inequalities as limit cases, and prove it for unconditional bodies, yielding a new proof of the volume entropy conjecture for the Hilbert metric for unconditional bodies. As a by-product, we obtain generalizations to higher moments of inequalities of Ball and Huang–Li, which in turn strengthen the Blaschke–Santaló inequality for unconditional bodies. Lastly, we introduce a regularization of the total volume of a smooth strictly convex $2$-dimensional set equipped with the Funk metric, resembling the O’Hara Möbius energy of a knot, and show that it is a projective invariant of the convex body.

We provide three $3$-dimensional characterizations of the $\mathbb{Z}$-slice genus of a knot, the minimal genus of a locally-flat surface in $4$-space cobounding the knot whose complement has cyclic fundamental group: in terms of balanced algebraic unknotting, in terms of Seifert surfaces, and in terms of presentation matrices of the Blanchfield pairing. This result generalizes to a knot in an integer homology $3$-sphere and surfaces in certain simply connected signature zero $4$-manifolds cobounding this homology sphere. Using the Blanchfield characterization, we obtain effective lower bounds for the $\mathbb{Z}$-slice genus from the linking pairing of the double branched covering of the knot. In contrast, we show that for odd primes $p$, the linking pairing on the first homology of the $p$-fold branched covering is determined up to isometry by the action of the deck transformation group on said first homology. As an application of the new upper and lower bounds, we complete the calculation of the $\mathbb{Z}$-slice genus for all prime knots with crossing number up to $12$.

The Carter tensor is a Killing tensor of the Kerr–Newman spacetime, and its existence implies the separability of the wave equation. Nevertheless, the Carter operator is known to commute with the D’Alembertian only in the case of a Ricci-flat metric. We show that, even though the Kerr–Newman spacetime satisfies the non-vacuum Einstein-Maxwell equations, its curvature and electromagnetic tensors satisfy peculiar properties which imply that the Carter operator still commutes with the wave equation. This feature allows to adapt to Kerr–Newman the physical-space analysis of the wave equation in Kerr by Andersson–Blue $\href{https://www.jstor.org/stable/24523010}{[4]}$, which avoids frequency decomposition of the solution by precisely making use of the commutation with the Carter operator. We also extend the mathematical framework of physical-space analysis to the case of the Einstein–Maxwell equations on Kerr–Newman spacetime, representing coupled electromagnetic-gravitational perturbations of the rotating charged black hole. The physical-space analysis is crucial in this setting as the coupling of spin‑1 and spin‑2 fields in the axially symmetric background prevents the separation in modes as observed by Chandrasekhar $\href{https://doi.org/10.1007/978-94-009-6469-3_2}{[19]}$, and therefore represents an important step towards an analytical proof of the stability of the Kerr–Newman black hole.

We establish a sharp upper-bound for the first non-zero eigenvalue corresponding to an even eigenfunction of the Hilbert–Brunn–Minkowski operator (the centro-affine Laplacian) associated to a strongly convex $C^2$-smooth origin-symmetric convex body $K$ in $\mathbb{R}^n$. Our isospectral inequality is centro-affine invariant, attaining equality if and only if $K$ is a (centered) ellipsoid; this is reminiscent of the (non affine invariant) classical Szegö–Weinberger isospectral inequality for the Neumann Laplacian. The new upperbound complements the conjectural lower-bound, which has been shown to be equivalent to the $\log$-Brunn–Minkowski inequality and is intimately related to the uniqueness question in the even $\log$-Minkowski problem. As applications, we obtain new strong non-uniqueness results for the even $L^p$-Minkowski problem in the subcritical range $-n \lt p \lt 0$, as well as new rigidity results for the critical exponent $p = -n$ and supercritical regime $p \lt -n$. In particular, we show that any $K$ as above, which is not an ellipsoid, is a witness to non-uniqueness in the even $L^p$-Minkowski problem for all $p \in (-n, p_K)$ and some $p_K \in (-n, 0)$, and that $K$ can be chosen so that $p_K$ is arbitrarily close to $0$.

The presented paper is a continuation of the series of papers [$\href{https://academic.oup.com/imrn/advance-article-pdf/doi/10.1093/imrn/rnz259/30788308/rnz259.pdf}{17}$, $\href{https://dx.doi.org/10.4310/CNTP.2020.v14.n4.a2}{18}$]. In this paper, utilizing Batyrev and Borisov’s duality construction on nef‑partitions, we generalize the recipe in [$\href{https://academic.oup.com/imrn/advance-article-pdf/doi/10.1093/imrn/rnz259/30788308/rnz259.pdf}{17}$, $\href{https://dx.doi.org/10.4310/CNTP.2020.v14.n4.a2}{18}$] to construct a pair of singular double cover Calabi–Yau varieties $(Y,Y^\vee)$ over toric manifolds and compute their topological Euler characteristics and Hodge numbers. In the $3$-dimensional cases, we show that $(Y, Y^\vee)$ forms a topological mirror pair, i.e., $h^{p,q} (Y ) = h^{3-p,q} (Y^\vee)$ for all $p, q$.