In light of recent advances in conformal blow-up methods for the positive mass theorem, including He--Shi--Yu, Bi--Hao--He--Shi--Zhu, and Brendle--Wang, we develop a Schoen--Yau type singular dimension descent method for positive scalar curvature obstructions in arbitrary dimensions. We prove obstructions to positive scalar curvature on enlargeable manifolds and establish the corresponding cubical width inequalities and two-systole estimates. The method also applies to enlargeable AM--PI spaces, giving a positive scalar curvature obstruction when the singular set has Assouad codimension greater than \(3-2/n\).

An elastic space curve is a critical point of the bending energy subject to appropriate constraints. An analytic representation, equivalent to the spherical pendulum equation, leads to an 11-parameter description of the space of 3D elastic curve segments. We give a numerically stable method for recovering the 11 parameters from a given elastic curve segment. Using this, we give a fast and stable method to approximate an arbitrary space curve segment by a 3D elastica. Applications include interactive design with exact elastic curves and CAD surface rationalization for robotic hot-blade cutting.

In this paper, based on recent results for the prescribed Hermitian-Yang-Mills (HYM) tensor and its twisted variants by Fan-Wang-Yang-Yau, we provide a dynamical construction of Hermitian-Einstein metrics on stable holomorphic vector bundles and its extension to Higgs bundles. Additionally, in the appendix, we use the heat flow method to give a new proof of the existence and uniqueness of solutions to the twisted prescribed HYM tensor equation, as well as its generalization to Higgs bundles.

Mean curvature flow is a fundamental geometric evolution equation in which a submanifold moves in the normal direction with velocity equal to its mean curvature vector. Self-shrinkers arise naturally as self-similar solutions to the mean curvature flow and play an important role as models for finite-time singularities. Among nontrivial examples of compact embedded self-shrinkers, the rotationally symmetric self-shrinking torus constructed by Angenent is one of the most important. However, the uniqueness of the Angenent torus remains a major open problem. In this paper, we study rotationally symmetric self-shrinkers of type $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$ from the point of view of ordinary differential equations. We analyze the profile curves of rotationally symmetric self-shrinkers, focusing on the behavior of their vertical points and the curves traced out by these points as the initial height varies. We give a new proof of the existence of the Angenent torus by showing that two families of vertical-point trajectories must intersect. We further derive the linearized equation associated with the rotationally symmetric self-shrinker equation and apply a Sturm-type comparison theorem to obtain sufficient conditions for the monotonicity of horizontal-point trajectories. In particular, we prove a comparison theorem for solutions near the spherical self-shrinker $x^{2}+r^{2}=2n$, and establish partial monotonicity results for the curves of horizontal points. These results provide a possible approach to the uniqueness problem for the Angenent torus.

We define a mass invariant adapted to the spectral scalar curvature for asymptotically hyperbolic 3-manifolds with toroidal infinity and show its positivity under a lower bound on the spectral scalar curvature. In addition, we show a rigidity theorem and some band width estimates under similar assumptions.

We prove that the top wedge power of the Ricci form of a complete non-compact Kähler manifold with positive sectional curvature has finite integral. Using a result of Chen-Zhu, an immediate consequence is the quasiprojectivity of such manifolds under the assumption of bounded sectional curvature. A key new idea to prove Bézout estimates along with a Lipschitz weight with finite Monge-Ampère mass is used in the proof of the main result.

Our goal is to survey the development of positive mass theorem and the Yamabe equation on CR manifolds in recent years. We introduce the notion of the mass in several complex variables or CR geometry. We then consider the Yamabe problem on CR manifolds to find a minimizer for the CR-Sobolev quotient. The positive mass theorem plays a key role in finding a solution to the Yamabe equation with minimum energy for the positive curvature case. We mainly focus on the team works in the following three papers [CMY17], [CMY23] and [CC22], on a positive mass theorem in 3-dimensional CR geometry, the CR-Sobolev quotient of Rossi spheres, and the 5-dimensional situation, respectively.

We obtain several constructions of uniformly positive scalar curvature complete Riemannian metrics on open manifolds. For dimension $n\geq3$, we show that if such a manifold admits a proper Morse function $f$ bounded below such that $f$ has no critical points of index $\geq n-2$, then it admits a uniformly positive scalar curvature metric. On the other hand if such a manifold admits a positive scalar curvature metric along with a compact exhaustion $\{U_i\}$ such that the boundary of each $U_i$ is minimal, then it also admits a uniformly positive scalar curvature metric. For dimension $4 \leq n\leq 7$, we show that if the manifold has product ends and a positive scalar curvature metric with $C$-quadratic decay at infinity for $C>4π^2$ with respect to some basepoint, then the existence of a mean convex hypersurface far enough from the basepoint implies the existence of a uniformly positive scalar curvature metric on the manifold. We study some applications of these results, including showing that if an open manifold of dimension $n\geq 3$ that admits no uniformly positive scalar curvature metric has a positive scalar curvature metric with mean convex exhaustion, then it admits a mean convex foliation of compact sets sufficiently close to the ends. On the other hand, if such a manifold has a mean concave exhaustion, then its ends admit a mean concave foliation.

We study Gromov's exact-lift two-form method in scalar-curvature geometry. For a closed Riemannian spin manifold carrying a homologically non-trivial closed two-form whose lift to the universal cover is exact, we prove the sharp hyperbolic scalar-curvature comparison with the bottom of the spectrum of the universal Riemannian covering. The two-form enters through Gromov's twisted \(L^2\)-index, which produces harmonic spinors for a family of small unitary twists. We analyze the equality case by interpreting the refined Kato equality defect conformally and use the harmonic spinors to construct a parallel spinor with respect to a suitable conformally related metric. This yields that the original metric is Einstein. In the positive-spectrum case, this method implies that the universal cover is real hyperbolic.

Poisson and Jacobi structures play a fundamental role in the geometric description of many systems arising in classical mechanics. In most cases, the corresponding bivector field is induced by a non-degenerate 2-covariant tensor. In this paper, we present a unified framework for constructing the associated brackets by studying the Poisson and Jacobi structures induced by these tensors. More specifically, under suitable assumptions on the tensor, we derive a formula for computing the Schouten-Nijenhuis bracket of the associated bivector field in terms of the curvature of a certain distribution and the exterior derivative of a differential form. This formula provides the obstruction to the existence of a Poisson or Jacobi structure. To illustrate the theory, we recover the classical brackets associated with symplectic, locally conformally symplectic, cosymplectic, and contact geometries. Finally, we characterize the conditions under which fat bundles and almost cosymplectic structures of order $p$ determine a Jacobi bracket.

Given a fixed past-directed timelike vector field, does there exist a time function whose gradient is optimally aligned with it? We address this question by introducing a functional that, on the one hand, captures the misalignment between the timelike vector field and the gradients of suitable Sobolev functions, and, on the other hand, penalizes null gradients. Our analysis focuses on compact subsets of smooth stably causal spacetimes. More precisely, we prove that, under suitable assumptions on the Sobolev index and the strength of the null gradient penalization, there exists a unique smooth temporal function which minimizes the considered functional. We refer to this minimizer as the \emph{alignment time function}. Furthermore, several useful properties of the alignment time function are established: there exists a canonical procedure to improve its steepness, it is stable under $C^{p}$ convergence of the underlying metrics and vector fields and it inherits the symmetries shared by the metric and the given vector field.

We place the attention token on the group: a token is an element $g_i$ of a matrix Lie group $G$ -- a bare transformation, with no feature payload and no external action $ρ(g)$ carrying it. To our knowledge this is the first attention construction whose tokens are bare matrix Lie group elements: their score is the closed-form algebra norm of the relative pose rather than a learned kernel, and it reaches the affine full-frame groups that every irrep- or surjective-exp-based method must exclude. We call it Lie-Algebra Attention. Once tokens are group elements, the rest follows with none of the usual representation-theoretic machinery. The relative geometry of a pair is canonical, $g_i^{-1} g_j$, so the pairwise invariant $w_{ij} = \log(g_i^{-1} g_j)$ is intrinsic rather than designed; equivariance under the diagonal $G$-action is tautological, and the cocycle condition holds automatically. The attention score is the negative squared algebra norm, $s_{ij} = -\|\log(g_i^{-1} g_j)\|_λ^2/τ$: the canonical proximity kernel under a block-weighted Frobenius inner product, with no irreducible representations, spherical harmonics, Clebsch-Gordan products, or learned kernel. The construction applies to any matrix Lie group on a chosen logarithm chart containing the relative poses, including the non-compact non-abelian affine groups with scale and shear that no vector-token attention method reaches: neither the irrep tradition nor surjective-exp methods. Three sequence-completion experiments, on SE(2), SO(3), and Aff(2), bear this out: the closed-form score matches a learned MLP kernel on the same invariant and outperforms it on SE(2), using 50 to 80x fewer score parameters, while a vector-token baseline breaks invariance by five to twelve orders of magnitude.

We develop a geometric framework for constrained optimization problems with inequality constraints through the introduction of quadratic slack variables. This formulation makes it possible to employ the language of Riemannian geometry and to solve the problem via the embedded gradient vector field method. We lift the feasible set to a smooth submanifold of an extended ambient space. The stratified structure of the resulting constraint manifold is analyzed in detail, yielding a natural partition according to which constraints are active. Using the embedded gradient vector field formalism, we derive explicit, determinantal formulas for the Lagrange multiplier functions directly from the geometry of the constraint manifold, recovering and re-framing the classical Karush-Kuhn-Tucker first-order necessary conditions without invoking the classical Lagrange multiplier method. Second-order optimality conditions are obtained by computing the restricted Hessian on each stratum, and a complete sign condition on the Lagrange multipliers is identified as the geometric counterpart of the classical complementary slackness condition. The theory is illustrated on the double ice-cream cone example, where the geometry of the problem determines the nature and number of local minima.

We prove that, in every degree, the rational Spin bordism classes represented by manifolds admitting metrics with positive Ricci curvature span exactly the kernel of the $\hat A$-genus. More precisely, for \[ R=Ω_*^{Spin}\otimes\mathbb{Q},\qquad J=\ker(\hat A:R\longrightarrow\mathbb{Q}[u]),\] the $\mathbb{Q}$-span of bordism classes of Ricci-positive Spin manifolds equals $J$ in each degree. This answers, in the differentiable rational Spin category, a question about rational bordism obstructions to positive Ricci curvature which was raised in the context of complex elliptic genera. The proof uses smooth complete intersections of an odd number $\ell$ of quadrics \[ Y_{m,\ell}\subset \mathbb{CP}^{2m+\ell}, \qquad \ell=1,\, 3,\, \ldots,\, 2m-1. \] These manifolds have real dimension $4m$, are Spin and Fano, and therefore admit metrics with positive Ricci curvature. A first-order thickening of the $\hat A$-genus induces $m-1$ linear functionals on $(J/J^2)_{4m}$. Their values on the classes $[Y_{m,\ell}]$ are governed by polynomials $P_{m,q}(\ell)$ of strictly increasing degrees $q+1=1$, $2$, $\ldots$, $m-1$. This gives full rank by a polynomial-interpolation argument.

We introduce a Dirac structure on the tangent bundle of a configuration manifold, called a \textit{Lagrange--Dirac structure}, which is naturally induced by the Lagrangian two-form associated with a (possibly degenerate) Lagrangian and a constraint distribution. This structure provides a unified geometric framework for Lagrange--Dirac dynamical systems, encompassing nonholonomic, degenerate Lagrangian, and symmetric systems. In the hyperregular case, the system recovers a first-order formulation of the Lagrange--d'Alembert equations. Although nonholonomic dynamics does not preserve the Lagrangian two-form, we show that the underlying Lagrange--Dirac structure is preserved up to gauge transformations, yielding a natural gauge covariance property. We also formulate an intrinsic variational principle directly on the tangent bundle, referred to as the \textit{Lagrange--d'Alembert--Dirac principle}, which recovers Hamilton's principle in the unconstrained case and the Lagrange--d'Alembert principle in the hyperregular constrained case. Furthermore, we develop a reduction theory for systems with Lie group symmetry, deriving a reduced Lagrange--Dirac structure over the Lie algebra that yields the Euler--Poincaré--Dirac equations and a corresponding reduced variational principle. Finally, we illustrate the theory through examples including charged particles, electric circuits, and systems with Lagrangians linear in velocity, and present an infinite-dimensional extension to ideal fluids that naturally incorporates the incompressibility constraint and recovers the Euler equations.

We establish the existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature using Perron-Frobenius theory. We establish a sharp upper bound for the largest eigenvalue of the associated Ricci matrix in terms of the maximum degree. Turning to structural properties, notably, the existence of a positive-curvature Einstein metric implies the tree must be a caterpillar. Furthermore, these metrics exhibit radial monotonicity, with edge weights decreasing strictly away from the maximal edge.

This paper shows that the concept of complex frequency, originally introduced to characterize the dynamics of signals with complex values, constitutes a generalization of eigenvalues when applied to the states of linear time-invariant (LTI) systems. Starting from the definition of geometric frequency, which provides a geometrical interpretation of frequency in electric circuits that admits a natural decomposition into symmetric and antisymmetric components associated with amplitude variation and rotational motion, respectively, we show that complex frequency arises as its restriction to the two-dimensional Euclidean plane. For LTI systems, it is shown that the complex frequencies computed from the system's states subject to a non-isometric transformation, coincide with the original system's eigenvalues. This equivalence is demonstrated for diagonalizable systems of any order. The paper provides a unified geometric interpretation of eigenvalues, bridging classical linear system theory with differential geometry of curves. The paper also highlights that this equivalence does not generally hold for nonlinear systems. On the other hand, the geometric frequency of the system can always be defined, providing a geometrical interpretation of the system flow. A variety of examples based on linear and nonlinear circuits illustrate the proposed framework.

We introduce large $p$ asymptotic geometric quantities associated with $p$-capacity, the first $p$-eigenvalue, and the Maz'ya constant on complete noncompact Riemannian manifolds. We prove the hierarchy $$ \mathcal{V}(M)\geq \mathcal C(Ω)\geq Λ(M)=\mathcal M(M)\geq0, $$ and show that, under a centered-ball isoperimetric condition or a rotational symmetry condition, these quantities coincide with the volume entropy or the dimension. In the Hadamard nonpositively curved case it also agrees with the topological entropy of the geodesic flow. As an application, combining with the entropy rigidity theorem, we obtain a characterization of hyperbolic manifolds. We also prove a second-order refinement. For a Hadamard manifold with compact quotient, under certain condition, the first-order large $p$ capacitary limit detects volume entropy, whereas the logarithmic second-order correction detects the rank.

We study a novel type of mechanical structures, composed of spatial four-bar linkages, that are bistable, that is, they allow for two distinct configurations. These structures have an interpretation as quad nets in the Study quadric which we use to prove existence of assemblies with an unbounded number of links and joints. We propose a purely geometric construction of such objects, starting from infinitesimally flexible quad nets in Euclidean space and applying Whiteley de-averaging. This point of view situates the problem within the broader framework of discrete differential geometry and enables the construction of bistable structures from well-known classes of quad nets, such as discrete minimal surfaces. In contrast to many other construction methods for bistable structures, our approach does not rely on numerical optimization and it allows for simple control of relevant geometric parameters such as axis positions and snap angles.

We study the free energy and dynamics of a closed elastic filament (a one-dimensional curve in two dimensions) coupled to a scalar concentration field representing, for example, an absorbed species. The density variable has a tendency to phase-separate whereas the local spontaneous curvature is concentration-dependent. We address analytically and by simulation both the free energy landscape and the dynamics (the latter comprising a coupled Willmore flow and Cahn--Hilliard gradient flow on the full differential geometry of a closed filament), addressing issues that previous work typically sidestepped by restricting to the Monge gauge. Specifically we find that the closure constraint for a deformable filament qualitatively changes the free energy landscape compared with either a rigid closed filament or an open elastic one, admitting metastable and stable states with more than one domain of each type. By numerical global free energy minimization we explore equilibrium morphologies across a wide range of model parameters. For selected parameter values we present fully dynamical results, tracking the time evolution of the various contributions to the free energy and confirming the emergence of both metastable and equilibrium multi-domain morphologies.

We give a representation of the extension class associated to a holomorphic fibration by curvature, generalizing the work of Atiyah on holomorphic principal bundles in a natural way. As an application, we obtain a nonlinear analogue of the classical result of Weil on characterizing the existence of flat connections on holomorphic vector bundles over compact Riemann surfaces. We further establish a faithful functor from the category of nonlinear flat bundles reductive of Kähler type to the category of nonlinear Higgs bundles over the same base, which is assumed to be a compact complex manifold of Kähler type. Finally, we establish a notion of nonlinear harmonic bundle and prove that the variation of nonabelian Hodge structure is a nonlinear harmonic bundle in the rank one case and in the semisimple case.

In this paper, we consider the Dirichlet problem for a class of prescribed Hessian quotient type curvature equations in Minkowski space. For non-convex domains, we prove the existence theorem by establishing the \emph{a priori} estimates without subsolution assumption and Serrin-type condition.

The goal of the paper is to introduce a convergence à la Gromov-Hausdorff for Lorentzian spaces, building on $ε$-nets consisting of causal diamonds and relying only on the time separation function. This yields a geometric notion of convergence, which can be applied to synthetic Lorentzian spaces (Lorentzian pre-length spaces) or smooth spacetimes. Among the main results, we prove a Lorentzian counterpart of the celebrated Gromov's pre-compactness theorem for metric spaces, where controlled covers by balls are replaced by controlled covers by diamonds. This yields a geometric pre-compactness result for classes of globally hyperbolic spacetimes, satisfying a uniform doubling property on Cauchy hypersurfaces and a suitable control on the causality, and a curvature-driven pre-compactness result. The final part of the paper establishes several applications: we show that Chruściel-Grant approximations are an instance of the Lorentzian Gromov-Hausdorff convergence here introduced, we prove that timelike sectional curvature bounds are stable under such a convergence, we introduce timelike blow-up tangents and discuss connections with the main conjecture of causal set theory.

In this paper we study the spectral analysis of Bochner-Kodaira Laplacians on an Abelian variety, complex projective space $\mathbb{P}^{n}$ and a Grassmannian with a holomorphic line bundle. By imitating the method of creation and annihilation operators in physics, we convert those eigensections (of the \textquotedblleft higher energy" level) into holomorphic sections (of the \textquotedblleft lowest energy" level). This enables us to endow these spectral bundles, which are defined over the dual Abelian variety, with natural holomorphic structure. Using this conversion expressed in a concrete way, all the higher eigensections are explicitly expressible using holomorphic sections formed by theta functions. Moreover, we give an explicit formula for the dimension of the space of higher-level eigensections on $\mathbb{P}^{n}$ through vanishing theorems and the Hirzebruch-Riemann-Roch theorem. These give a theoretical study related to some problems newly discussed by string theorists using numerical analysis. Some partial results on Grassmannians are proved and some directions for future research are indicated.

This is the third work in a series on the (local in time) well-posedness of the initial boundary value problem (IBVP) for the vacuum Einstein equations in general relativity with geometric boundary conditions. Here we study the conformal-mean curvature boundary conditions, consisting of the conformal class of the boundary metric and mean curvature of the boundary. We prove that at metrics of uniformly bounded geometry to all orders, the linearized problem has a solution space with dense range in $C^{\infty}$ and establish a Holmgren-type uniqueness theorem valid for general smooth linearized solutions. These results require the addition of an arbitrary corner angle term at the intersection of the Cauchy surface and the timelike boundary.

In this paper we consider sequences $u_j:B_2\subseteq M\to N$ of stationary harmonic maps between smooth Riemannian manifolds with uniformly bounded energy $E[u_j]\equiv \int |\nabla u_j|^2\leq Λ$ . After passing to a subsequence it is known one can limit $u_j\to u:B_1\to N$ with the associated defect measure $|\nabla u_j|^2 dv_g \to |\nabla u|^2dv_g+ν$, where $ν= e(x)\, H^{m-2}_S$ is an $m-2$ rectifiable measure \cite{lin_stat}. For a.e. $x\in S=\operatorname{supp}(ν)$ one can produce a finite number of bubble maps $b_j:S^2\to N$ by blowing up the sequence $u_j$ near $x$. We prove the energy identity in this paper. Namely, we have at a.e. $x\in S$ that $e(x)=\sum_j E[b_j]$ for a complete set of such bubbles. That is, the energy density of the defect measure $ν$ is precisely the sum of the energies of the bubbling maps.

A decorated surface S is an oriented topological surface with marked points on the boundary considered modulo the isotopy. We consider the moduli space of hyperbolic structures on S with geodesic boundary, such that the hyperbolic structure near each marked point is a cusp, equipped with a horocycle. This space carries a volume form. Let us fix the set K of distances between the horocycles at the adjacent cusps, and the set L of lengths of boundary circles without cusps. We get a subspace M(S; K,L) with the induced volume form Vol(K,L). However, if the cusps are present, the volume of the space M(S; K,L) is infinite. We introduce the exponential volume form exp(-W)Vol(K,L), where W is a positive function on the moduli space, given by the sum over cusps of the hyperbolic areas enclosed between the cusp and the horocycle at the cusp. We prove that the exponential volume, defined as the integral of the exponential volume form over the moduli space M(S; K,L), is always finite. We suggest that the moduli spaces M(S; K,L) with the exponential volume forms are the true analogs of the classical moduli spaces of Riemann surfaces, with the Weil-Petersson volume forms. In particular, they should be relevant to the open string theory. We support this by proving an unfolding formula for the integrals of measurable functions multiplied by the exponential volume form. It expresses them as finite sums of similar integrals over moduli spaces for simpler surfaces. They generalise Mirzakhani's recursions for the volumes of moduli spaces of hyperbolic surfaces. We show that exponential volumes for elementary decorated surfaces give rise to a commutative algebra E, which we call the positive Hecke-Whittaker algebra for PGL(2,R). Exponential volumes for all decorated surfaces and unfolding formulas extend the algebra E to all decorated surfaces.

We study the systole of a model of random hyperbolic 3-manifolds introduced by Petri and Raimbault, answering a question posed in that same article. These are compact manifolds with boundary constructed by randomly gluing truncated tetrahedra along their faces. We prove that the limit, as the volume tends to infinity, of the expected value of their systole exists and we give a closed formula of it. Moreover, we compute a numerical approximation of this value.

We present general reduction procedures for Courant, Dirac and generalized complex structures, in particular when a group of symmetries is acting. We do so by taking the graded symplectic viewpoint on Courant algebroids and carrying out graded symplectic reduction, both in the coisotropic and hamiltonian settings. Specializing the latter to the exact case, we recover in a systematic way the reduction schemes of Bursztyn-Cavalcanti-Gualtieri.

The K-energy functional is extended to complexified Kähler classes, providing a variational approach to study the scalar curvature equation with B-field introduced by Schlitzer and Stoppa. The extended K-energy is convex along geodesics in the space of almost calibrated representatives of the complexified Kähler class. This fact is used to show that, in some situations, solutions of the scalar curvature equation with B-field are unique in their class, up to pullbacks by reduced automorphisms of the manifold.