We continue our local singularity analysis for Ricci flow initiated in arXiv:2006.16227. Building on that framework, we study Type I singular points in general Ricci flows, without assuming any global Type I curvature bound, and prove that the scalar curvature must blow up at a Type I rate at each such point in all dimensions. As a consequence, Ricci flows with bounded scalar curvature cannot develop Type I singular points. This extends earlier results of the first author with Enders and Topping and with Mantegazza that relied on a global Type I assumption. We then adapt the same local perspective to ancient Ricci flows and analyse the curvature behaviour as time goes to negative infinity, showing in particular that every ancient Type I point exhibits scalar curvature behaviour of ancient Type I order.

For any closed surface of genus at least $3$, equipped with any bounding spin structure, we show that the index difference, viewed as a map from the fundamental group of the space of Dirac-invertible Riemannian metrics to $\KO^{-4}(*)$, is non-trivial. For products of two such surfaces, equipped with any spin structure, we prove that the corresponding space of Dirac-invertible Riemannian metrics is not contractible. We discuss the relationship of this result to the existence of metrics with harmonic spinors in dimension $4$.

Given a finite collection of Lagrangian submanifolds $\mathscr L$ in a compact symplectic manifold $X$, we construct a cyclic, filtered, strictly unital curved $A_{\infty}$ category $\mathcal L$ and develop Floer theory of closed-open maps and open-closed maps. Using them, we prove that, whenever the map from the quantum cohomology of $X$ to the Hochschild cohomology of the Fukaya category $\mathcal L$ with objects $\mathscr L$ is injective, the following consequences follow: (1) any other Lagrangian submanifold equipped with a weak bounding cochain lies in the category split-generated by $\mathscr L$, and (2) the Hochschild homology and cohomology of the Fukaya category are isomorphic to quantum cohomology. In the exact case a similar result was obtained in [Ab]. We also provide some applications.

We prove Sobolev-type inequality and local volume noncollapsing with optimal exponents for compact Kähler manifolds of uniformly bounded $q$-Nash entropy.

We revisit flows of tensorial $H$-structures for closed and connected Lie subgroups $H\leqslant\mathrm{SO}(n)$, focusing on the topology of isometric classes. We prove that the natural map assigning to an $H$-structure its induced Riemannian metric is surjective and satisfies a parametric homotopy lifting property. Since the space of Riemannian metrics is contractible, the full space of $H$-structures is homotopy equivalent to any fixed isometric class. For parallelizable manifolds, especially flat tori, these classes reduce to mapping spaces into $\mathrm{SO}(n)/H$. We discuss almost Hermitian, $\mathrm{SU}(m)$, $\mathrm{G}_2$, and $\mathrm{Spin}(7)$ structures on flat tori, showing that their isometric classes and moduli modulo orientation-preserving diffeomorphisms may have infinitely many connected components. We relate this topology to the variational theory of the intrinsic torsion energy. On the unrestricted space of $H$-structures, the functional is scale-degenerate in dimensions $n>2$: its infimum is zero on every nonempty path component, and its only critical points are torsion-free structures. Inside fixed isometric classes this homothetic escape direction is absent. We reinterpret finite-time singularity formation as concentration in nontrivial isometric homotopy classes with zero energy infimum, and contrast this with cohomological classes, such as $\mathrm{U}(3)$-structures on the flat $6$-torus, which have positive lower bounds and admit smooth harmonic representatives from holomorphic maps into $\mathbb{CP}^3$. Finally, we revisit analytical aspects of our earlier work: we prove a lifting principle for metric-dependent flows, reinterpret the Ricci $H$-flow, derive a general evolution identity for isometric flows, and extend the harmonic-flow theory beyond the original structural assumptions.

This manuscript investigates the spectral geometry of Riemannian submersions whose fibers have a basic mean curvature. By restricting the Laplace--Beltrami operator to the space of basic functions, we reduce the spectral problem on $M$ to the spectral problem for a weighted Laplacian on the base manifold, where the weight is determined by the fiber-volume function $S$. We derive a summation formula for the reciprocal of the basic Dirichlet eigenvalues (Basel-type series). Furthermore, using the framework of Supersymmetric Quantum Mechanics (SUSYQM), we establish a supersym\-me\-tric duality relating the basic Dirichlet and Neumann spectra under the trans\-for\-ma\-tion $S \mapsto 1/S$.

We study spherical T-duality for iterated sphere bundles. We show that for a class of iterated sphere bundles the cohomological data contained in its Gysin sequences can be repackaged into data for a vanishing Massey product. We further show that if these bundles are endowed with an integral cohomology class of transgressive degree one, then they have a T-dual iterated sphere bundle, namely, the one associated to the same Massey product read backwards.

We prove rigidity in the equality case of the sharp bottom spectrum estimate under scalar curvature lower bound. Under the same topological assumptions as in our previous work, a closed manifold $(M,g)$ with $\mathrm{Sc}_g\geq -n(n-1)$ and $\lambda_1(\widetilde M,\widetilde g)=(n-1)^2/4$ must be hyperbolic. This gives rigidity results for closed hyperbolic manifolds and for closed manifolds admitting a metric of nonpositive sectional curvature.

We establish a complete structural classification for continuum-wise hyperbolic surface homeomorphisms. Specifically, we prove that a surface homeomorphism is cw$_F$-hyperbolic if, and only if, it is a pseudo-Anosov homeomorphism whose singularities consist exclusively of spines (1-prongs). Furthermore, we classify these systems up to topological conjugacy, showing that every such homeomorphism is conjugate to either an Anosov automorphism on the torus $\mathbb{T}^2$ or to its standard hyperelliptic quotient on the sphere $\mathbb{S}^2$. As a rigid consequence of this classification, we show that such dynamics are strictly obstructed on surfaces of genus greater than one, the Klein bottle, and the projective plane.

We prove a singularity theorem in which the classical focusing hypothesis of Hawking--Penrose theory is replaced by a condition on asymptotic volume growth. Under the strong energy condition, we introduce asymptotic volume-expansion invariants associated with a compact Cauchy hypersurface and show that a uniform positive lower bound on these invariants implies past timelike geodesic incompleteness. More precisely, we obtain an explicit upper bound on the time-separation from the hypersurface to its chronological past. The theorem extends to globally hyperbolic Lorentzian length spaces satisfying the synthetic strong energy condition $\mathsf{TCD}^e_p(0,N)$, yielding an inextendibility result valid without any smoothness or differentiability assumption. We also prove an area comparison theorem for equidistant hypersurfaces and a volume singularity theorem based on related asymptotic expansion invariants.

We prove sub-Riemannian Selberg trace formulae for compact quotients of SL(2, R). Using the Fourier decomposition along the SO(2)-fibers, we reduce the heat trace computation to the Selberg trace formula for Maass Laplacians on the hyperbolic plane. The resulting formula has an identity contribution and a hyperbolic contribution, the latter involving a character-dependent theta factor over closed geodesics. We then use this trace formula to compute the zeta-regularized determinant of the sub-Laplacian. The determinant formula is remarkably compact and is expressed in terms of a determinant depending only on the base hyperbolic surface and an explicit relative Selberg product.

Originally motivated by creating first-person computer visualizations within Riemannian manifolds -- the author was led to study deformable-body mechanics, as rigid-body mechanics is not available in a generic Riemannian manifold due to its lack of nontrivial isometry group. Hyperelasticity is a particularly nice sub-category of continuum mechanics in which a deformable, elastic body's behavior is determined by a stored energy density function. This allows problems to be posed variationally, and powerful tools brought to bear on studying and solving them. This article presents numerical simulations of static solutions to a particular class of problems in hyperelastic mechanics in 2-dimensional Riemannian manifolds in which a flat hyperelastic body $B$ is embedded into a region $\Omega$ in a nowhere-flat surface $S$ of revolution $z=z\left(r\right)$ such that $\left|K\left(r\right)\right|$ decreases as $r\to\infty$, where $K$ denotes the Gaussian curvature of $S$. For example, the funnel $z=-r^{-1}$ or the paraboloid $z=\frac{1}{2}r^{2}$. Because $B$ is flat, the body can't achieve a zero-stored-energy configuration, and restorative forces arise in the body to move it toward a region of lower stored energy -- meaning, toward a flatter configuration. With the addition of a gravitational potential $U\left(r\right)=z\left(r\right)$ on $S$, forces act on the body to pull it toward $r=0$. If the body has sufficient stiffness and remains within the region $\Omega$, then the body has an equilibrium configuration in which the body's deformation-response forces perfectly cancel the gravitational forces. Such a configuration represents a kind of "levitation" phenomenon within this surface. The numerical implementation of this problem will be detailed and the resulting numerical solutions and various consequences discussed.

In this paper, we investigate finite and infinite decomposition formulas for zero mean curvature (ZMC) graphs in three-dimensional spaces with different metrics, including Euclidean space, Lorentz--Minkowski space, and isotropic 3-space. We first derive new Euler--Ramanujan-type identities yielding decompositions for the conjugate of Scherk's first minimal surface in terms of dilated catenoids. We then employ Weierstrass factorisation and power series techniques to obtain infinite decompositions for a broad class of ZMC graphs in isotropic 3-space into helicoids, logarithmoids of revolution, and Enneper surfaces. We further extend these decomposition results to larger families of ZMC surfaces across these spaces by considering surfaces arising from the López--Ross transformation, Bonnet rotation, and a one-parameter family of metric deformations. We also investigate finite decompositions, establishing finite analogues of Scherk tower decompositions in both Euclidean and isotropic settings. In addition, we prove a theorem characterising finite decompositions of isotropic minimal surfaces. Finally, we discuss applications of the resulting decomposition theory to lamellar structures.

We study discrete analogues of classical spectral geometric inequalities and extremal eigenvalue problems on graphs. The well-known Krahn--Szegő inequality states that the minimum of $\lambda_2(\Omega)$ among bounded open sets of $\mathbb{R}^n$ with given volume is achieved by the union of two identical balls $\mathbb{R}^n$. Firstly, we establish a Krahn--Szegő type inequality for trees. For trees with a fixed number of interior vertices and boundary leaves, we completely characterize the extremal structures that minimize the second Dirichlet eigenvalue. Secondly, we develop a nodal domain method for adjacency matrices. By proving a nodal domain theorem in adjacency version for graphs, we obtain upper bounds for the second largest adjacency eigenvalue $\rho_2(G)$ of $G$ in given graph classes. These bounds imply some previous results. Finally, we settle the Aouchiche--Hansen conjecture (2010) on the second largest eigenvalue with given number of edges and clique number. We prove that for connected graphs $G$ of odd order $n \geq 5$, $|\rho_2| \cdot \omega \leq m-2$, with equality if and only if $G$ consists of two complete graphs of orders $\frac{n+1}{2}$ and $\frac{n-1}{2}$ joined by an edge or a path. For even $n \geq 2$, the quantity $|\rho_2| \cdot \omega - m$ is maximized exactly when $G$ is the join of two copies of $K_{n/2}$ by an edge. The core of the methods developed in this paper is to regard a connected graph as an internally disconnected graph with Dirichlet boundary condition. This perspective allows us to transfer nodal domain techniques from continuous spectral geometry to discrete settings and to obtain sharp extremal characterizations across diverse graph classes.

Let $\Sigma$ be a closed embedded minimal hypersurface in $\mathbb{S}^{n+1}$. We establish an improved lower bound for the first non-zero eigenvalue of the induced Laplace-Beltrami operator on $\Sigma$. It is slightly better than the bound of Duncan-Sire-Spruck.

We introduce the calibration energy for oriented immersions into Euclidean space, quantifying the deviation from calibrated geometry. A key property is that this energy may remain finite for infinite-volume immersions, while a null-Lagrangian structure ensures that it has the same first variation as the volume functional. We establish an exact dissipation identity for the calibration energy along oriented, proper mean curvature flows in arbitrary dimensions and codimensions, under a mild local-volume bound. This provides a new, finite variational framework for mean curvature flow beyond the finite-volume setting. Our result yields several applications, including rigidity for solitons and convergence for two-dimensional immortal solutions. In particular, every proper self-expander with finite constant-coefficient calibration energy must be a plane in all dimensions and codimensions.

A recent conjecture by Joel Fine posits a relationship between the coefficients of the HOMFLY polynomial of a knot $K$ in the 3-sphere $S^3$, and the signed count of minimal surfaces in hyperbolic 4-space $\mathrm{H}^4$ meeting the sphere at infinity at $K$, with prescribed genus and self-intersection number. In this paper, we develop a novel machine learning framework based on Physics-Informed Neural Networks (PINNs) to solve the minimal surface equation in hyperbolic space. We utilise this framework to test Fine's Conjecture by constructing near-minimal surfaces bounding various families of knots in $S^3$. Furthermore, we develop an algorithmic method to find self-intersections and compute their sign. For every knot analysed, the computationally discovered minimal surfaces and their self-intersection numbers perfectly align with the predictions of Fine's Conjecture, providing empirical evidence for it.

Cosymplectic and normal almost contact structures are analogues of symplectic and complex structures that can be defined on 3-manifolds. Their existence imposes strong topological constraints. Generalized geometry offers a natural common generalization of these two structures: $B_3$-generalized complex structures. We prove that any closed orientable 3-manifold admits such a structure, which can be chosen to be stable, that is, generically cosymplectic up to generalized diffeomorphism.

We prove the weak cosmic censorship conjecture in $2+1$ spacetime dimensions for the circularly symmetric Einstein-scalar field system in the presence of a negative cosmological constant $\Lambda<0$. More precisely, we show that for any integer $k\geq2$, the maximal development of generic $C^k$ initial data does not contain naked singularities. An essential step of the proof is establishing the presence of a mass gap. In particular, this implies that all naked singularities have infinite blueshift, which represents the main instability mechanism.

In this paper, we give a new proof of the splitting theorem on manifolds with nonnegative spectral Ricci curvature proved in [APX24, CMMR24, HW26]. Furthermore, by constructing weighted minimizing geodesics at infinity, we show that minimal hypersurfaces with finite index in manifolds with nonnegative biRic curvature must have finite ends, generalizing the result of Li-Wang [LW04] on manifolds with nonnegative sectional curvature.

With the goal of solving optimisation problems on non-Riemannian manifolds, such as geometrical surfaces with sharp edges, we develop and prove the convergence of a forward-backward method in Alexandrov spaces with curvature bounded both from above and from below. This bilateral boundedness is crucial for the availability of both the gradient and proximal steps, instead of just one or the other. We numerically demonstrate the behaviour of the proposed method on simple geometrical surfaces in $\mathbb{R}^3$.

Carrollian manifolds offer an intrinsic geometric framework for the physics in the ultra-relativistic limit. The recently introduced Carrollian Lie algebroids are generalised to the setting of $\rho$-commutative geometry, (also known as almost commutative geometry), where the underlying algebras commute up to a numerical factor. Via $\rho$-Lie-Rinehart pairs, it is shown that the foundational tenets of Carrollian geometry have analogous statements in the almost commutative world. We explicitly build two toy examples: we equip the extended quantum plane and the noncommutative $2$-torus with Carrollian structures. This opens up the rigorous study of noncommutative Carrollian geometry via almost commutative geometry.

Developments in Carrollian gravity and holography necessitate the use of singular Carroll vector fields, a feature that cannot be accommodated within standard Carrollian geometry. We introduce Carrollian Lie algebroids as a framework to study such singular Carrollian geometries. In this approach, we define the Carroll distribution as the image of the kernel of the degenerate metric under the anchor map. The Carroll distribution is, in general, a singular Stefan--Sussmann distribution that will fluctuate between rank-1 and rank-0, and so captures the notion of a singular Carroll vector field. As an example, we show that an invariant Carrollian structure on a principal bundle leads to a Carrollian structure on the associated Atiyah algebroid that will, in general, have a singular Carroll distribution. Mixed null-spacelike hypersurfaces, under some simplifying assumptions, also lead to examples of Carrollian Lie algebroids. Furthermore, we establish the existence of compatible connections on Carrollian Lie algebroids, and as a direct consequence, we conclude that Carrollian manifolds can always be equipped with compatible affine connections.

We argue that modular classes of Q-manifolds provide an efficient method for addressing the existence of supersymmetric Berezin volumes in the supergeometric representation theory of the $\mathcal{N}=2$ $d=1$ supertranslation algebra. We establish a cohomological coherence criterion for the existence of a Berezin volume that is invariant under both of the supercharges.

We investigate the $h$-principle problem for fat distributions. These are maximally non-integrable distributions with natural symplectisations and contactisations, that generalize contact distributions to higher corank. We focus on the corank-$2$ case, where we study a natural class of submanifolds, which we call prelegendrians. Their key feature is that they admit a canonical Legendrian lift to the contactisation. Our main results state that the $h$-principle fails for these submanifolds in all dimensions. To the best of our knowledge, this is the first example of rigidity in the study of maximally non-integrable distributions, outside of contact topology. First, we find an infinite family of $(2n+1)$-tori in the standard fat $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$, with the following two properties: (1) They all represent the same formal prelegendrian class, (2) but they are not prelegendrian isotopic because they are distinguished by pseudoholomorphic curve invariants of their Legendrian lift. Secondly, we define the notion of prelegendrian stabilization in $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$. This allows us to take an arbitrary prelegendrian and produce another one, in the same formal class, whose Legendrian lift is loose. In order to prove these results we also develop the fundamentals of the theory of prelegendrians. This includes: (1) introducing the notion of front projection in $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$, (2) proving that pseudoholomorphic curve invariants are robust under perturbations of the fat structure, allowing us to transport our results to non-standard fat structures, (3) introducing a zooming argument showing that any fat structure in dimension $6$ admits prelegendrians.

A vector field $V$ on any (semi-)Riemannian manifold is said to be mixed Killing if for some nonzero smooth function $f$, it satisfies $L_VL_Vg=fL_Vg$, where $L_V$ is the Lie derivative along $V$. This class of vector fields, as a generalization of Killing vector fields, not only identify the isometries of the manifolds, but broadly also contain the class of homothety transformations. We prove an essential curvature identity along those fields on any (semi-)Riemannian manifold and thus generalize the Bochner's theorem for Killing vector fields in this setting. Later we study it in the framework of almost coKähler structure and we prove that the Reeb vector field $\xi$ on an almost coKähler manifold is mixed Killing if and only if the operator $h=0$. Moving further, we completely classify almost coKähler manifolds with $\xi$ mixed Killing vector field in dimension 3. In particular, if $\xi$ on an $\eta$-Einstein almost coKähler manifold is mixed Killing, then the manifold is of constant scalar curvature with $h=0$. Also we show that on any $(\kappa,\mu)$-almost coKähler manifold, $\xi$ is mixed Killing if and only if the manifold is coKähler. In the end we present few model examples in this context.

We prove that the conjectured capillary Blaschke-Santaló inequality holds for any unconditional, strictly convex capillary hypersurface when $θ\in \left(0, \tfracπ{2}\right)$. Moreover, for $θ\in \left(\tfracπ{2}, π\right)$, we show that the capillary volume product has no finite upper bound.

The existence of small-time asymptotics for the heat kernel of the Kohn Laplacian on a general CR manifold has remained an open problem. In this paper, we resolve the problem in the non-degenerate case. More precisely, let $X$ be a compact oriented CR manifold of dimension $2n+1$, $n \ge 1$, with a nondegenerate Levi form of constant signature $(n_-, n_+)$. Suppose that condition $Y(q)$ holds at each point of $X$, we establish the small-time asymptotics of the heat kernel of Kohn Laplacian. Suppose that condition $Y(q)$ fails, we establish the small-time asymptotics of the kernel of the difference of the heat operator and Szegő projector. As an application we define the analytic torsion on compact oriented nondegenerate CR manifolds and study its dependence on changes of the metrics. Let $L^k$ be the $k$-th power of a CR complex line bundle $L$ over $X$. We establish the asymptotics, as $k \to \infty$, of the analytic torsion with values in $L^k$, under a variant of spectral gap condition. Furthermore, when $X$ admits a transversal CR $S^1$-action, we establish the small-time asymptotics of the $S^1$-equivariant heat kernel of the Kohn Laplacian with values in $L^k$. As an application we define the $S^1$-equivariant Quillen metric with values in $L^k$ and study its dependence on changes of the metrics. Finally, we establish the asymptotics, as $k \to \infty$, of the $S^1$-equivariant analytic torsion with values in $L^k$.

We propose an approach to Carrollian geometry using principal $\mathbb{R}^\times$-bundles ($\mathbb{R}^\times:= \matthbb{R} \setminus \{0\}$) equipped with a degenerate metric whose kernel is the module of vertical vector fields. The constructions allow for non-trivial bundles, and a large class of Carrollian manifolds can be analysed in this formalism. A key result in this is that once a principal connection has been selected, there is a canonical non-degenerate metric that can be leveraged to circumvent the difficulties associated with a degenerate metric. Within this framework, we examine the Levi-Civita connection and null geodesics.

This paper investigates the differential-geometric and topological properties of the Cayley condition in Poncelet porism for triangles, defined as the locus of pairs of non-degenerate conics that admit a Poncelet triangle. While the algebraic condition for this porism, established by Cayley, is classical, the geometric nature of the set of solutions has remained largely unexplored. We demonstrate that this Cayley set is a smooth, connected, 9-dimensional complex manifold. This is proven by showing it is an open subset of a smooth algebraic variety endowed with a trivial fiber bundle structure over the space of non-degenerate conics. To further analyze its structure, we construct the moduli space of transversely intersecting conic pairs under the action of $\mathbb{P}GL_3(\mathbb{C})$ and identify it with an open subset of $\mathbb{CP}^2/S_3$. We compute the fundamental group of a generic orbit. The elliptic j-invariant is then introduced as a holomorphic map on this space of conics, which factors through this moduli space. We analyze the subset of the Cayley set where this map is a submersion - its regular part - which corresponds to excluding points whose j-invariant is one of the critical values $0$ or $1728$. We prove that this regular part is the total space of a fiber bundle over $\mathbb{C} \setminus \{0,1728\}$. This structure allows for the computation of the fundamental group via the long exact sequence of homotopy. Finally, we provide a principal bundle formulation for the Poncelet correspondence itself over orbits of conic pairs.