Papers

We establish Quillen equivalences between successive truncations of multicomplexes for the model structures that detect weak equivalences simultaneously on prescribed pages of the two associated spectral sequences. In particular, the natural comparison from $4$-multicomplexes to bicomplexes, which is relevant to the comparison between almost complex and complex geometry, is a Quillen equivalence. This answers a question of Cirici, Livernet, and Whitehouse.

We build an unbiased Monte Carlo estimator of the integral of any $C^1$ function on a prequantized compact symplectic manifold against a smooth Riemannian volume form, taking for quadrature nodes the determinantal point process associated with an appropriate spectral projection of the Bochner-Schrödinger operator. We show that the estimator satisfies a central limit theorem, and the decay rate of the mean squared error reaches the optimal worst-case rate investigated by Bakhvalov in Euclidean spaces. These results extend previous results of Lemoine and Bardenet on Monte Carlo methods on compact complex manifolds.

We prove that the uniform doubling property holds for left-invariant Riemannian metrics on every step-two Carnot group and the sharp uniform doubling constant is $2^Q$, where $Q$ is the homogeneous dimension. More generally, every such metric satisfies the Bishop–Gromov type estimate.

We establish the rigidity statement in the equality case of the hyperspherical-radius inequality of Brendle, Tsiamis, and Wang for compact spin fill-ins with scalar curvature bounded below. More precisely, let $(M^{n\geq 3},g)$ be a compact, connected Riemannian spin manifold having a connected boundary $\Sigma$ and scalar curvature satisfying $\mathrm{scal}_g\geq -n(n-1)$. We prove that equality in the upper bound \[ \inf_{\Sigma}H\leq (n-1)\sqrt{1+\operatorname{Rad}(\Sigma)^{-2}} \] given by Brendle, Tsiamis, and Wang holds if and only if $(M,g)$ is isometric to a geodesic ball in hyperbolic space.

In this largely expository paper we provide a deep and explicit exploration and exposition of the tangent structure on the category of schemes $\mathbf{Sch}_{/S}$ whose tangent functor $T(X) = T_{X/S}$ is the relative tangent scheme of Grothendieck described in \emphÉléments de Géométrie Algébrique} 4. In particular we provide explicit descriptions of the ways that the bifibration of quasicoherent sheaves and bifbration of quesicoherent sheaves of algebras over schemes may be built from the ways in which the bifibrations of modules and commutative algebras over commutative rings interact. We also show the ways in which these interactions give rise to an explicit description of the standard tangent structure on the category of schemes in terms of sheaves of Kähler differentials, properties of the relative spectrum functor, and more. Finally, we show that quasi-coherent sheaves can be reconstructed from their category of differential bundles by showing that for quasi-separated schemes $X$ and $Y$, there is an isomorphism $X \cong Y$ if and only if there is an equivalence of categories $\mathbf{DBun}(X) \simeq \mathbf{DBun}(Y)$.

We introduce and investigate a notion analogous to stable causality for Lorentzian foliations, considerably extending the framework of transverse causality initiated in a recent work. It is well-known that in spacetime geometry stable causality is characterized by several equivalent conditions, such as the stability of non-existence of causal loops under metric perturbations, the existence of smooth temporal functions, and relation-theoretic formulations via the so-called Seifert and $K^+$ relations. We define natural transversal analogues of these distinct formulations and establish partial logical implications between them in the foliation setting. Whether and to what extent the full equivalences can be established remains an open question for general foliations, partly due to the eventual non-Hausdorff nature and other topological complexities of arbitrary leaf spaces. Furthermore, we demonstrate that for the important class of simple foliations, which are defined by certain submersions, the equivalences of almost all the transverse analogues of stable causality are indeed recovered.

In this paper, we address the following question: if a flat torus $\mathbb{T}^n$ is isometrically and minimally embedded into a sphere $\mathbb{S}^N$, must its translation group extend to the isometry group of the ambient sphere? As shown by Robert Bryant, for $n=2$ the answer is positive. Furthermore, while Ying Lu, Peng Wang, and Zhenxiao Xie recently demonstrated that the answer is negative for immersions when $n \geq 3$, the question for embeddings remained open. This problem is deeply tied to the work of Mikhail Gromov and Anton Petrunin concerning optimal curvature bounds. Petrunin proved that any immersion of a torus into a unit ball must have a maximum normal curvature of at least $\sqrt{\frac{3n}{n+2}}$. This bound is attained, for example, by families of tori constructed by Gromov. We call the tori that attain this optimal bound "Gromov tori". In this work, we first demonstrate that any Gromov torus is intrinsically flat, lies within a sphere, and is minimal inside it. We then establish the necessary and sufficient conditions for defining these tori. Finally, we present our main result: for dimensions $n \ge 3$, there exists a non-equivariant embedded Gromov torus, which provides a definitive negative answer to the question above.

This paper considers planar mathematical billiards with smooth billiard tables. We prove that through any point outside a convex smooth caustic, there passes a unique billiard table (Lemma 2). Furthermore, we describe a billiard table having a nephroid as its caustic.

We study the problem of lifting global and local quaternionic torus actions to principal quaternionic torus bundles. Let \(Q^k=(\operatorname{Sp}(1))^k\cong (S^3)^n\), let \(G\) be a compact Lie group acting on a connected, locally finite CW complex \(X\), and let \(Q^k\longrightarrow P\longrightarrow X\) be a principal \(Q^k\)-bundle. We first formulate a quaternionic analogue of the obstruction-theoretic framework of Hattori–Yoshida. The existence of a lifted \(G\)-action implies that the isomorphism class of \(P\) lies in the image of the restriction map induced by the Borel construction \(X_G=EG\times_GX\). In particular, the second Chern class admits an equivariant extension. Once a continuous pseudo-lift has been chosen, its failure to define a genuine action is measured by a factor set with values in the generally nonabelian gauge group \(\mathcal G(P)\cong\Gamma(\operatorname{Ad}(P))\). We obtain a necessary and sufficient lifting criterion in terms of the trivializability of this factor set, and show that, once a single lift exists, the set of all lifts modulo gauge conjugacy is classified by a pointed nonabelian \(H^1\)-set. We then apply this global theory to local quaternionic torus actions. Pulling back to the universal covering of the orbit space, untwists a local \(Q^n\)-action and produces a globally defined action on the pulled-back manifold. A preliminary lift of this global action need not be compatible with the deck transformations. We define a gauge-valued nonabelian descent defect, establish its crossed-cocycle identities and transformation law, and prove that the original local action lifts if and only if the global lifting obstruction vanishes and the descent defect is trivializable. In the abelian case, these constructions reduce to the classical obstruction theory for lifts of local torus actions.

Let $M_s^n$ be a non-degenerate pseudo-Riemannian submanifold of a pseudo-Euclidean space with indefinite induced metric. Restricting the contact condition to spacelike or to timelike tangent directions gives two a priori different one-sided contact numbers. We prove that, for every prescribed order, ordinary, spacelike and timelike contact are equivalent; consequently, the three contact numbers coincide. The proof is inductive. A one-sided contact hypothesis first yields higher-order orthogonality identities for the iterated covariant derivatives of the second fundamental form. Their diagonal scalar expressions are then extended from either unit pseudo-sphere to the whole tangent space by a polynomial argument, and a second induction proves causal invariance. We also construct an explicit Lorentzian surface in $\mathbb E_4^8$ whose ordinary, spacelike and timelike contact numbers are all equal to six, and place the infinite-contact limit in the semi-Riemannian theory of geodesic normal sections and helical immersions.

Following Brendle's smooth approximation approach, we give a proof of Gromov's dihedral rigidity conjecture for convex polytopes.

In this note, we prove that, on weighted graphs, the Lin–Lu–Yau curvature coincides with the $p$-Ollivier curvature up to scaling whenever the idleness parameter $p\geq 1/2$. Moreover, the threshold $1/2$ is sharp. This extends an earlier result of Bourne et al. (Ollivier–Ricci idleness functions of graphs, SIAM J. Discrete Math., 32 (2018), no. 2, 1408-1424), where combinatorial graphs were considered. This observation yields a simple proof for the global existence and uniqueness of solutions of the Lin–Lu–Yau curvature flow in Bai et al. (Ollivier Ricci-flow on weighted graphs, Amer. J. Math. 146 (2024), 1723-1747).

We study the linearisation of formal vector fields and homotopy algebras using graded coalgebras. We prove that a formal vector field is linearisable if and only if it satisfies a splitting property, and we give an explicit recursive construction of a linearising coordinate transformation. This provides a direct coalgebraic proof of Bandiera's characterisation of linearisable, equivalently homotopy abelian, $L_{\infty}[1]$ algebras. We establish an associative analogue, characterising linearisability of $A_{\infty}[1]$ algebras by a splitting property for one-sided Hochschild cochains. We also examine the corresponding splitting property for standard Hochschild cochains and show that it is equivalent to $A_{\infty}[1]$ commutativity in the sense of Briggs and Gélinas. Finally, we establish a corresponding criterion for formal endomorphisms and give sufficient conditions for splitting that recover Basto-Gonçalves' linearisation theorem for admissible formal vector fields.

In this note, we show that a weak mean curvature flow with critical forcing term obtained by Liu–Tonegawa (2024) satisfies the BV-type area change formula, that is, their flow is not only a Brakke flow but also a generalized BV flow, which is proposed by Stuvard–Tonegawa (2024). To establish this result, we identify minimal conditions under which a Brakke flow satisfies the area-change formula. As an application of our main theorem, we derive a lower bound for the extinction time of generalized BV flows with a critical forcing term. We also outline the proof of a compactness theorem for generalized BV flows.

A long-standing conjecture in Hermitian geometry says that a compact Hermitian manifold with constant Chern holomorphic sectional curvature $c$ is Kähler for $c\neq 0$ and Chern flat for $c=0$. Although the conjecture has been established in complex dimension two, it remains open in general in higher dimensions. We verify the conjecture for compact balanced threefolds when $c\leq 0$. For compact locally conformally Kähler manifolds, Chen, Chen, and Nie established the case $c\leq 0$, while Huang and Wan recently settled the remaining case. Inspired by the approach of Huang and Wan, we investigate a generalization of the conjecture for canonical metric connections and establish it for connected compact locally conformally Kähler manifolds.

For maps from surfaces, harmonicity is conformally invariant and a conformal immersion is harmonic precisely when its image is minimal. Biharmonicity is a fourth-order extension of this theory, but it is not conformally invariant. A nonminimal immersion may therefore become biharmonic after a suitable change of the domain metric. In a three-dimensional space form, Ou's formulation reduces this problem to two coupled equations for the weighted mean curvature $U=\lambda^2H$: one scalar equation and one tangential equation. We ask whether these two equations can be organized as a single Dirac-type equation and what geometry is compatible with a first-order factorization. Restricting an ambient Killing spinor to the surface, we construct a natural Laplace-type operator $\mathscr B_c$ and prove that $\mathscr B_c(U\psi)=0$ is exactly equivalent to Ou's system. We then classify every factorization of $\mathscr B_c$ in the monic scalar–chiral class $(D+a+b\omega)(D+p+q\omega)$. In nonzero curvature, every such factorization is automatically mean-curvature-normalized and exists locally if and only if the surface has locally constant principal curvatures. The same rigidity holds for the normalized Euclidean branch; the remaining Euclidean factors form an exceptional holomorphic–antiholomorphic family characterized, away from planar points, by harmonicity of $\log(|A|^2-H^2)$. The rigidity mechanism is governed by a Dirac discriminant $9c-4|A|^2$. Its hypothetical non-CMC real branch reduces to a spherical Gauss–Codazzi system whose exact Frobenius torsion is strictly negative. Model examples finally show that factorization of the geometric operator is distinct from the existence of a positive conformal mode. Thus the scalar–chiral channel is complete but too rigid to generate new non-CMC examples, providing a precise baseline for broader Clifford-valued constructions.

Building on a commutative geometric algebra developed for 2D and 3D fields of complex vectors, this paper develops a vector framework for the differential geometry of compact Hermitian manifolds embedded in an ambient three-dimensional field of complex vectors. The underlying algebraic framework is constructed from the commutative geometric product of complex vectors, yielding a unified commutative algebra in both the 2D and 3D settings together with the corresponding vector differential and integral identities. These algebraic results provide the intrinsic foundation for a differential geometric formulation that is developed independently of the classical scalar coordinate approach. Within this framework, compact Hermitian manifolds are described entirely in vector form. Fundamental tensor relations governing the intrinsic and extrinsic geometry of complex manifolds are derived in vector form. The resulting framework establishes a unified algebraic and geometric setting in which the differential geometry of complex manifolds is developed directly from the commutative algebra of complex vectors.

Let $\varphi:\Sigma^2\looparrowright\mathbb{RP}^3$ be a closed immersed surface, with no orientability or, equivalently, two-sidedness assumptions. We establish the sharp quantitative estimate $$ \lambda_2(\Delta+|\sigma|^2+2)\leq2-\frac{2}{\operatorname{Area}(\Sigma)}\int_\Sigma H^2d\Sigma+\frac{4\pi\chi(\Sigma)}{\operatorname{Area}(\Sigma)}. $$ Our main contribution is the analysis of the borderline case, which rules out equality when $\chi(\Sigma)\leq0$. More precisely, $$ \lambda_2(\Delta+|\sigma|^2+2)<2\quad\text{whenever}\quad\chi(\Sigma)\leq0. $$ The proof combines the canonical Veronese embedding of $\mathbb{RP}^3$ into $\mathbb{S}^8$, the conformal test function method, an Obata-type rigidity argument, and the classification of closed flat minimal surfaces in $\mathbb{RP}^3$.

We investigate the deformable hypersurfaces of the conformally flat Riemannian products $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$, $2\leq k\leq n-1$, of hyperbolic space and a sphere with constant sectional curvatures $-1$ and $1$, respectively.

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$, $n\ge2$, and let $\alpha_\gamma(n)$ be the largest number for which \[ \gamma_n(\lambda K+(1-\lambda)L)^{\alpha_\gamma(n)} \ge \lambda\gamma_n(K)^{\alpha_\gamma(n)} +(1-\lambda)\gamma_n(L)^{\alpha_\gamma(n)} \] holds for all convex bodies $K,L\subset\mathbb{R}^n$ containing the origin and all $\lambda\in[0,1]$. In this paper, we prove that \[ \alpha_\gamma(n) =1-\frac{2}{n-1} \frac{\Gamma(\frac n2)^2}{\Gamma(\frac{n-1}{2})^2}. \] The core of the proof is a raywise radial–tangential localization of the Hessian energy of a solution of a Neumann problem, which reduces source selection of the Neumann problem to a one-dimensional optimization. Monotonicity in the segment length and Laguerre spectral analysis determine the sharp one-dimensional value, whereas the planar endpoint is treated separately.

We study ruled minimal surfaces in a family of three-dimensional solvable Lie groups depending on a real parameter $p$ and endowed with left-invariant metrics. For $p\neq 0$, we classify all orthogonal ruled surfaces and construct new examples of minimal surfaces. For $p=0$, we parametrize all ruled surfaces and obtain additional examples of minimal surfaces.

We study large mass monopoles with structure group $\mathrm{SU}(2)$ or $\mathrm{SO}(3)$ on asymptotically conical $\mathrm{G}_2$-manifolds and Calabi–Yau $3$-folds, with fixed asymptotic class. After placing the AC asymptotic theory, the variational compactness theory of Parise–Pigati–Stern, and Li's singular abelian compactness theory in a common $\Theta$-monopole framework, we prove that the mass-renormalized Yang–Mills–Higgs and intermediate energy measures converge to $8\pi\|T\|$ for a compactly supported calibrated integral codimension-three cycle $T$. This identifies the two limiting currents and shows that the variational calibration inequalities are saturated. Using this common limit as the starting point for a finer analysis, if $\mathcal S$ is the calibrated support, $\mathcal Z$ the Kuratowski upper limit of the Higgs zero sets, and $\mathcal C$ the limiting nonabelian locus, defined as the Kuratowski upper limit of Li's curvature concentration loci, then $\mathcal S\subset\mathcal Z\subset\mathcal C=\mathcal S\cup\mathcal O$, where $\mathcal O$ is precisely the obstruction to effective codimension-three monotonicity. For the cohomogeneity-one large mass families on the Bryant–Salamon $\mathrm{G}_2$-manifolds and the Stenzel Calabi–Yau $3$-fold, we prove that $\mathcal O=\varnothing$. On $X\setminus\mathcal C$ the sequence abelianizes; corrected longitudinal curvatures converge smoothly, and the remaining compactness alternatives are governed by $L^2$-harmonic $2$-forms.

We prove a Heintze–Karcher type inequality for a large class of log-convex weights in Euclidean and Hyperbolic spaces. As a consequence we obtain an Alexandrov theorem for $\lambda$-self expanders.

We first construct the differential refinement of singular homology, verifying that it satisfies the axioms dual to their cohomological counterparts. Then, we define the differential cap product, leading to Poincaré duality. The essential uniqueness of both differential homology and the cap product is proven. Moreover, we construct the relative, non-compact, and local coefficient versions, so that we can state Poincaré and Lefschetz dualities for every smooth manifold with our without boundary. Afterwards, we develop the analogous refinement of any rationally-even homology theory, with the same properties. We also define a "differentiation map" for fibre bundles, dual to the integration map in cohomology. We conclude by sketching some possible applications of this theory in mathematical physics.

We construct random open covers of higher-rank locally symmetric spaces using a construction we call scaffolded Poisson processes. Let $X=G/K$ be a symmetric space of noncompact type and real rank at least $2$. We prove a general vanishing theorem for the normalized torsion in first homology along sequences of torsion-free lattices in $G$. In particular, if $G$ is simple, we get \[ \dfrac{\log |H_1(M_n;\mathbb{Z})_{\operatorname{tors}}|}{\mathrm{vol}(M_n)} \longrightarrow 0 \] for any sequence of distinct manifolds $M_n = \Gamma_n \backslash X$. This answers a question of Abért, Gelander, and Nikolov, and confirms the degree-one vanishing with trivial integral coefficients predicted by a conjecture of Bergeron and Venkatesh in the higher-rank setting. In addition, we get quantitative bounds with respect to the minimal injectivity radius for both the torsion in first homology and the minimal number of generators of $\Gamma$. Finally, we prove the analogous statements for affine buildings.

We develop a new analytical framework for the local deformation theory of compact cohomogeneity-one Einstein metrics. The framework combines an intrinsic reformulation of the Einstein boundary-value problem with an auxiliary theory of equivariant harmonic maps and leads to what we call the Einstein Detection Principle. The auxiliary harmonic maps generate a rich variational, elliptic and spectral framework while introducing no additional Einstein degrees of freedom.

We provide a new approach to the classification, up to equivalence, of abelian complex structures on nilpotent Lie algebras. As an application, we recover the already known classifications in dimensions 4 and 6 and provide a classification in dimension 8 for 1-abelian complex structures. Some results about the general case are given.

We study local minimizers of a sextic-potential Landau–de Gennes energy for nematic liquid crystals in the small-elastic-constant limit. Under the uniform energy and $L^\infty$ bounds, these minimizers converge to a locally energy-minimizing harmonic map $\mathbf{Q}_0$ into a biaxial vacuum manifold. The main result of this paper is that such a limiting map $\mathbf{Q}_0$ has no interior point singularities. The proof relies on a geometric identification of the lifted Frobenius metric on the universal cover $\mathbb{S}^3$ with a rescaled Berger metric. For a hypothetical tangent cone at a point singularity, its link is a nonconstant harmonic two-sphere into the Berger sphere. We construct a smooth variation field adapted to the Hopf direction and show that it induces a quantitative instability estimate, in contradiction with the shifted stability inequality inherited from local minimality. This replaces the usual round-sphere test fields by a target-specific construction and rules out interior point defects in the biaxial setting. The result is sharp in view of the well-known existence of point defects in the uniaxial theory.

Grayson and Gage proved that an immortal curve shortening flow of simple closed curves on a closed surface converges subsequentially to a closed geodesic, and they asked whether this limiting geodesic is unique. We answer this question negatively by constructing a smooth Riemannian metric on $\mathbb{S}^2$ and an immortal simple closed curve shortening flow that converges along different sequences of times to every geodesic in a one-parameter family of distinct simple closed geodesics.

Let \(X\) be a closed Riemann surface and let \(q\in H^0(X,K^4)\) be a nonzero holomorphic quartic differential on \(X\). For \(t >0\), the ray \(tq\) determines a family of Hitchin representations in the \(\operatorname{PSp}(4,\mathbb R)\)-Hitchin component. We study, as \(t\to+\infty\), the asymptotic behavior of their holonomy along closed curves. We obtain explicit asymptotic formulas for all singular values and for the absolute values of all eigenvalues of the holonomy. Their logarithmic growth rates are given by integrating the local fourth roots of \(q\) along the saddle connections forming the geodesic representative of the curve with respect to the singular flat metric \(\lvert q\rvert^{1/2}\). No restriction is imposed on the orders of the zeros of \(q\).