Papers

We classify left-invariant Rarita-Schwinger fields on 2- and 3-dimensional Lie groups with left-invariant Riemannian metrics. The left-invariant Rarita-Schwinger equation is reduced to an algebraic system on the associated metric Lie algebra. We show that, in dimension 2, non-trivial examples occur only in the flat abelian case. In dimension 3, apart from the flat abelian case, the only non-trivial examples occur on SU(2) equipped with a special left-invariant metric, for which the space of left-invariant Rarita-Schwinger fields has complex dimension 2. This metric is a Berger metric on $S^3$.

Let $M$ be a closed Kähler surface. We prove that every Riemannian metric $g$ on $M$ with $\operatorname{Sc}_g\geq-\lambda^2$, where $\lambda\geq 0$, satisfies $$ \lVert M\rVert\leq \frac{27}{2}\,\lambda^4\operatorname{vol}_g(M). $$ This proves Gromov's quantitative scalar-curvature–simplicial-volume conjecture for closed Kähler surfaces. We also construct infinitely many non-Kähler symplectic 4-manifolds of general type with positive simplicial volume for which the same estimate holds.

This paper establishes sharp upper bounds for $p$-capacities $\mathrm{Cap}_{1<p<\infty}$ in the hyperbolic space $\mathbb{H}^n$ through hyperbolic quermassintegrals and effective curvature radii. The quermassintegral comparisons involve $W_{n-1}$, $W_{k+1}+k(n+1-k)^{-1}W_{k-1}$, and the pair $W_1\mid W_2$. For star-shaped, mean-convex or h-convex hypersurfaces, inverse mean curvature flow further produces curvature radii determined by $L^q$-averages of the normalized mean curvature and by moments of its squared hyperbolic excess. These radii convert the resulting estimates into sharp geodesic-ball comparisons for the capacity-to-area ratio. In the range $p>2m+1$, an interpolating radius combines the $2m$-th curvature-excess radius with the $L^\infty$ curvature scale, thereby linking the finite-moment and supremum regimes. Equality in the sharp comparisons characterizes geodesic balls.

This paper presents an innovative approach to the capacities-to-masses for the Laplace-Beltrami operators on the complete Riemannian manifolds, unexpectedly solving the open problem posed within \cite[Remark 2.16]{GPS} on the limiting variational capacity.

We show that, for Weil–Petersson random closed hyperbolic surfaces of large genus, the normalized weighted count of prime geodesics with norm in the interval $(X,X+H]$ is asymptotically Gaussian, provided $H \leq X$ and $H/\log X\to\infty$ as $X\to\infty$. In particular, it applies to intervals which are not necessarily short.

In this work, we introduce the Koszul solitons of the Hesse-Koszul flow, i.e., self-similar solutions for a geometric flow on Hessian manifolds. We obtain some partial differential equations and basic properties of Koszul solitons, and provide some examples. The key idea is to investigate the relationship between affine manifolds and complex manifolds, this method can also be used to study some other geometric problems on Hessian manifolds. It is worth noting that Hessian manifolds, as a special case of affine manifolds, have wide applications in statistics and information geometry.

Following previous work on metric bisectors with respect to the Korányi metric, we study here bisectors in the Heisenberg group $\mathfrak{H}$ equipped with the Carnot–Carathéodory metric $d_{cc}$.

Existence of canonical metric on a canonical model of projective singular variety was a long standing conjecture and the major part of this conjecture is about varieties which do not have definite first Chern class (most of the varieties do not have definite first Chern class). There is a program which is known as Song-Tian program for finding canonical metric on canonical model of a projective variety by using Minimal Model Program. In this paper, we apply Song-Tian program for mildly singular pair $(X;D)$ via Log Minimal Model Program where $D$ is a simple normal crossing divisor on $X$ with conic singularities. We show that there is a unique $C^\infty$-fiberwise conical Kähler-Einstein metric on $(X;D)$ with vanishing Lelong number which is twisted by logarithmic Weil-Petersson metric and an additional term of Fujino-Mori [72] as soon as we have fiberwise KE-stability or Kawamata's condition of Theorems 2.28, or 2.30(in $C^0$ case). In final we highlight that how the complete answer of this question can be reduced to CMA equation corresponding to fiberwise Calabi-Yau foliation(due to H.Tsuji) which is still open.

We study resolution-independent relative logarithmic source complexes for AKSZ-BV-BFV theory on face-oriented Joyce manifolds with generalized corners. Under the geometric hypotheses (G1)-(G3), the Dupont-Panzer-Pym total relative logarithmic complexes of admitted smooth monoidal resolutions represent $j_!\Omega_{X^\circ}^\bullet$ and carry the same compactly supported derived trace. The comparison is deliberately unfiltered: it identifies total relative cocycle classes through the common interior, but not individual resolved faces, separate BFV descendants, nonlinear mapping spaces, or absolute regularized integrals. For the intrinsic $b$-source ${}^{b}T[1]X$ we formulate the additional Stokes and transfer data needed to recover strict facewise structures. A multiplicative regularized Stokes trace system on Joyce's face category gives the presymplectic intrinsic BV-BFV identity and incidence descent. The finite datum (G4) is a linear subdivision/aggregation transfer criterion; (G5) is a separate cyclic field-level criterion for classical abelian BF theory. Neither is a general existence theorem, and no general strict multiplicative intrinsic trace is constructed here. We prove the interval contractions needed for codimension-two collars and analyze the positive real conifold through the common star refinement of its two diagonal resolutions. On an explicit finite product-Whitney logarithmic class the exceptional square satisfies (G4) after normal-face totalization, and the transferred differential is exactly the signed intrinsic incidence differential. Its finite algebraic BF dual gives a cyclic logarithmic-cellular shadow, but not the full continuum intrinsic $b$-de Rham datum (G5). General nonlinear continuum pushforward, existence of the intrinsic multiplicative trace, and loop-level logarithmic graph integrals remain open.

Motivated by the geometric properties of curved tubes $T\left(P,a\right)$ defined by closed curves $P$, we compute the Cheeger constant $h\left(T\left(P,a\right)\right)$ in the real space forms of arbitrary dimensions with constant sectional curvature. The structure and properties of the system of Fermi coordinates allows us to parametrize the curved tube and straightforwardly compute the upper bound of $h\left(T\left(P,a\right)\right)$ using the exact formulas for the area and volume of $T\left(P,a\right)$. Next, we derive the lower bound by combining the geometry of tubes with a calibration-type argument. The key idea is to describe the tube using geodesic spheres moving along the underlying curve $P$, which provides a natural outward direction and makes the estimate geometrically transparent. This allows us to compute the lower bound via the divergence theorem. Finally, for the class of unbounded curved tubes in noncompact real space forms, we also compute the Cheeger constant and prove that there is no finite-volume Cheeger set.

For integers $n\ge 1$, consider assertions: $\mathbf{(P_n)}$: Let $\Omega \subset S^n$ be a spherical domain enclosed by totally geodesic $S^{n-1}$'s with non-obtuse dihedral angles. Then in $[0,2(n+1)]$, its Neumann spectrum can only take values among $\{0,n,2(n+1)\}$. Moreover, $n$ is a Neumann eigenvalue if and only if the corresponding eigenfunction is the restriction of a linear function in $\mathbb{R}^{n+1}$, while $2(n+2)$ is a Neumann eigenvalue if and only if the corresponding eigenfunction is restriction of a quadratic polynomial in $\mathbb{R}^{n+1}$. $\mathbf{(Q_n)}$: A weak solution $u$ to $\Delta u = f$ with the Neumann boundary condition, with $f$ Hölder continuous, in a conical polyhedral domain $\Omega$ in $\mathbb{R}^n$ with non-obtuse dihedral angles, is in $C^{2,\alpha}_{loc}(\overline{\Omega})$. We prove the implications \[\mathbf{(Q_n)} \Rightarrow \mathbf{(P_n)},\qquad \mathbf{(P_n)}\Rightarrow \mathbf{(Q_{n+1})}.\] Consequently, both assertions hold in all dimensions. These give the optimal Neumann eigenvalue lower bound and $C^2$ elliptic regularity in non-obtuse Riemannian polyhedral domains.

The only Einstein metrics currently known on complex projective spaces are homogeneous: the Fubini-Study metric, arising via the Hopf fibration; and, in odd complex dimensions, Ziller's metric, obtained as a canonical variation along the twistor fibration over the quaternionic projective space. In 1965, Berger proved that the Fubini-Study metric is the unique Kähler-Einstein metric on the complex projective space, and posed the question of whether other Einstein metrics exist. We construct the first inhomogeneous Einstein metrics on complex projective spaces of complex dimension $n$ for $3\leq n \leq 7$, answering Berger's question affirmatively for the complex even dimensional cases $n=4$ and $n=6$.

There are several non-equivalent notions of infinitesimal symmetry in the literature of second order linear differential equations: Lie point symmetries, vertical (gauge) symmetries, operator symmetries, infinitesimal contact symmetries, and Lie–Bäcklund operators. We construct an explicit correspondence among the first three, We then describe the Lie algebra $\mathcal L_\Omega(\mathbb U)$ of infinitesimal contact transformations of the contact system $\Omega=\langle dy-y'\,dx\rangle$, with coefficients in a differential field $\U$ of functions of $x$ and $y$. We obtain a canonical decomposition $\mathcal L_\Omega(\mathbb U)=\prod_{k\ge0}\mathcal L_\Omega^k\mathbb U$ into $\mathbb C$-vector spaces, each parametrized by $\mathbb U$ (by $\mathbb U\oplus\mathbb U$ for $k=0$), and we compute the algebraic differential formulae for the Lie bracket in these coordinates. Applied to the symmetry problem, we prove that a contact vector field with generating function $W$ is a symmetry of if and only if $A^{2}W=aW+b\,AW$, where $A$ is the vector field in the jet space corresponding to the equation; equivalently, if and only if $W=F_1(u_1,u_2)\phi_1+F_2(u_1,u_2)\phi_2$ for arbitrary functions $F_1,F_2$ of the two first integrals of $A$ and a basis $\phi_1,\phi_2$ of solutions. The symmetry algebra is always parametrized by two arbitrary functions of two variables. It also shows that the decomposition of $\Lom(\U)$ never captures the whole symmetry algebra: for $a\neq0$ the graded part reduces to the point symmetries, while for $y''=0$ it is an infinite dimensional but still proper subspace, and in neither case is it a Lie subalgebra. Finally we make precise the transformation law $W\mapsto\mu^{-1}(W\circ\varphi)$ for characteristics under a contact transformation with conformal factor $\mu$, which governs the transport of evolutionary representatives.

The aim of this manuscript is to show that the group of autoequivalences of an exact Courant algebroid over a compact base manifold is a tame Fréchet Lie group. Moreover, we compute its Lie algebra. Furthermore, we show that the space of generalized almost complex structures is a tame Fréchet manifold and that the canonical action of the group of autoequivalences on the space of generalized almost complex structures is smooth tame.

We study higher-order global estimates for the heat equation on Riemannian manifolds, both for static metrics and for metrics evolving under the Ricci flow. Under minimal geometric assumptions, we derive first-order regularizing estimates for log-solutions of the heat equation, together with upper second-order bounds with explicit constants. Our quantitative approach is based on integral duality methods proposed by L.\ C.\ Evans, J.-M.\ Lasry and P.-L.\ Lions in different settings.

We formulate the inverse problem in information geometry within a two-point tensorial framework and solve it for general metric-affine manifolds, without imposing any curvature or torsion constraints. The construction is explicit and starts directly from the given geometric data: the metric tensor is paired to the affine structure, realized through a local parallelism obtained from parallel transport. This yields a contrast bi-form inducing the original metric-affine manifold. The inverse problems for statistical manifolds admitting torsion and for statistical manifolds are then recovered by homotopical reduction. In this way, suitable pre-contrast and contrast functions are obtained, including several established constructions for statistical manifolds and SMATs. We apply the general theory to reductive homogeneous pseudo-Riemannian manifolds endowed with invariant affine connections. Particular attention is devoted to semisimple Lie groups with Cartan-Schouten connections and to odd-dimensional spheres equipped with Berger metrics.

Atiyah's Minkowski-space version of the configuration-of-points construction assigns to an admissible marked configuration of $n$ worldlines a collection of $n$ binary forms of degree $n-1$, whose roots are the ordered retarded celestial directions. He conjectured that these forms are always linearly independent. We disprove this conjecture for every $n\ge3$. For $n=3$, an explicit planar one-parameter family yields a real coefficient determinant with exactly one simple zero in a specified interval. At this parameter, all six ordered celestial roots are distinct and the coefficient matrix has rank exactly two. A null-translation construction then multiplies the first three forms by a common factor and produces counterexamples for every $n>3$. Consequently, within the class of complete pairwise disjoint timelike affine lines, universal independence holds at $n=2$ and fails for every $n\ge3$; for each of the counterexamples, the normalized Atiyah–Sutcliffe determinant is defined and vanishes.

Let $k\in\mathbb N$ satisfy $1<k<n/2$, and let $\Sigma^p\subset\Sn^n$ be a closed smooth embedded submanifold. We prove that a complete conformal metric $g=v^{-2}g_{\Sn^n}$ on $\Sn^n\setminus\Sigma$ satisfying $\lambda(g^{-1}A_g)\in\Gk$ and $\sigma_k(g^{-1}A_g)=\kappa>0$ must obey \[ p\leq p_k(n), \] where $p_k$ is the model threshold determined by $\Hh^{p+1}\times\Sn^{n-p-1}$. When $k=2$ and $n=m^2$, we construct a smooth complete equality example on $\Sn^n\setminus\Sn^{(m^2-m-2)/2}$. We also prove that the strict inequality $p<p_k(n)$ holds under a finite positive linear-contact hypothesis.

We present a systematic study on the role of invariant connections in geometric mechanics. We first develop a comprehensive theory of reduction under left- and right-invariant connections on Lie groups, showing that the Euler–Poincaré and Lie–Poisson equations are independent of the connection. We then introduce a novel connection-dependent variational principle, where the Lagrangian depends on the velocity parallel-transported back to the initial point of the curve. For Cartan–Schouten connections, this leads to an integro-differential Euler–Poincaré equation that exhibits two distinct sources of nonlocality in time: a path-dependent term encoded in the parallel transport and a future-dependent term arising from a curvature integral. We reformulate this equation as a two-point boundary value problem and present a two-level numerical scheme. The general theory is illustrated on the Heisenberg group, where the equations simplify due to nilpotency, and on the rotation group, where the full integro-differential structure is retained.

We construct $U(1)$-symmetric $SU(2)$ instantons on the Taub-NUT space with generic holonomy, and instanton number and magnetic charge $(k,m)=(2,0)$, $(1,1)$, and $(2,-1)$. We construct these instantons via the bow construction, a Nahm-like non-linear transformation which identifies them with a set of bow data solving an integrable ODE and a series of algebraic relations on an interval. We show how the bow data recovers the data for all corresponding $U(1)$-symmetric instantons on flat $\mathbb{R}^4$ and calorons on $\mathbb{R}^3\times S^1$ in various limits, and we demonstrate the existence of Taub-NUT instantons that do not limit to $\mathbb{R}^4$ instantons nor calorons.

We investigate a compact Einstein-type manifold whose potential vector field generates a Riemannian foliation. In particular, we prove necessary conditions for such a manifold to be taut and to have a splitting property. Additionally, some properties of taut Riemannian foliations on a compact almost Ricci solitons and a compact Einstein manifolds are provided.

To combat oversmoothing in Graph Convolutional Networks, Sheaf Neural Networks (SNNs) were proposed as a generalization by equipping the graph with a sheaf structure and replacing the graph Laplacian with a sheaf Laplacian $\mathcal{L}$. Existing analyses connect sheaf diffusion to oversmoothing via the harmonic space ($\ker\mathcal{L}$), taking its absolute dimension as an indicator of anti-oversmoothing capacity. However, absolute dimension alone is not a reliable measure: certain sheaf configurations inflate $\dim \ker \mathcal{L}$ while their harmonic sections remain entirely constant, without enriching discriminative capacity. We instead introduce the first relative, geometric approach, yielding a precise characterisation of anti-oversmoothing capacity. Under natural conditions on stalk transportation and global sheaf structure, we establish an index-theoretic comparison criterion showing that one sheaf's harmonic space genuinely contains another's beyond trivial inflation. We illustrate this with a concrete instance and further introduce GyroSheaf, a sheaf with curved gyrovector-space stalks, extending the criterion to the non-linear setting via local tangent-space linearization. Experiments across ten models confirm the theoretical criterion: sheaf models violating the criterion collapse despite possessing index jumps, while compliant models maintain depth-stable representations.

We prove a sharp lower bound for the first nonzero Neumann eigenvalue of geodesic triangles of given diameter in two-dimensional space forms. The bound is given by the first positive radial Neumann eigenvalue of an one dimensional model; it is approached by degenerating isosceles triangles. When \(K>0\) and \(D=\pi/(2\sqrt K)\), equality is attained precisely by birectangular triangles. We also prove a hot-spots theorem for non-acute spherical triangles of diameter at most \(\pi/2\), and establish antisymmetry and eigenvalue monotonicity for isosceles spherical triangles of diameter \(\pi/2\).

Let $F:M^n\to\Sn^{n+q}(1)$ be a closed minimal immersion in the unit sphere, with $n\ge3$, $q\ge2$. Let $S$ be the squared length of its second fundamental form. We prove that if $M$ is not totally geodesic, then $$ \max_M S> \frac{n(288\sqrt6\,n-288)}{(288\sqrt6+192)n-137} >\frac{2n}{3}+\frac{31-9\sqrt6}{75}(n-2). $$

In the paper, we develop a structure theory for metric measure spaces with synthetic lower Ricci curvature bounds, known as RCD spaces. Under a positive injectivity-radius assumption, we recover a smooth differential structure and prove a regularity result: such spaces are $W^{1,p}_{\mathrm{loc}}\cap C^{0,\alpha}_{\mathrm{loc}}$-Riemannian manifolds whose weight functions also belong to $W^{1,p}_{\mathrm{loc}}\cap C^{0,\alpha}_{\mathrm{loc}}$, in both distance and harmonic charts for all $p<\infty$ and $\alpha \in (0,1)$. We also establish a quantitative lower bound for the harmonic radius, together with compactness theorems and quantitative geometric bounds. As a byproduct, we develop a comprehensive elliptic regularity theory in this setting. These results apply, in particular, to smooth weighted Riemannian manifolds and to RCD spaces satisfying a synthetic curvature upper bound, or CBA condition. Even in these settings, the resulting regularity statements are new. In the latter case, both the Riemannian metric and the weight function are shown to be locally Lipschitz. As further applications, we establish fibration theorems and use metric smoothing to confirm, in a synthetic framework, a conjecture of V. Kapovitch concerning almost flat manifolds with mixed curvature bounds.

In \cite {ChenFengLiXiXu2026}, a reverse projection inequality for general convex bodies was established in three dimensions. In this paper, we completely characterize the equality cases of this inequality. We further show that the reverse projection inequality fails in every dimension $n\geq9$. The counterexamples are constructed as full-dimensional polytopes with at most $n+2$ facets.

We prove that the volume of a compact connected Kähler manifold with holomorphic sectional curvature at least 2 is bounded above by the volume of the Fubini-Study metric of constant holomorphic sectional curvature 2 on the complex projective space of the same dimension. Moreover equality holds if and only if the manifold is biholomorphically isometric to complex projective space. This answers a question posed by Xiong and Yang. Our approach also yields a different proof of Zhang's sharp volume estimate and Liu's rigidity theorem for compact Kähler manifolds with positive Ricci curvature. In fact, our main result states that the same sharp volume estimate holds under a new curvature positivity condition (mean RC curvature positivity), which is implied by both positive Ricci curvature and positive holomorphic sectional curvature. The definition of this condition was inspired by the work of Yang. The proofs in this paper are due to ChatGPT 5.6 Sol Pro, and the paper is merely an exposition of its output. The proofs has been verified by the authors and they take full responsibility for any errors.

The symmetric space $X_n={\rm SL}(n,\Rb)/{\rm SO}(n)$ can be interpreted as the Teichmüller space of marked, unit volume, flat $n$-dimensional tori. It comes with a unique (up to scale) ${\rm SL}(n,\Rb)$-invariant metric $d_{X_n}$. In 1939 Teichmüller gave a modular interpretation of $d_{X_2}$ (the hyperbolic metric) in terms of an extremal mapping problem for quasiconformal dilatation. Such a modular interpretation for $d_{X_n}$ for $n\geq 3$ has remained unaddressed: the natural candidates - minimal quasiconformal dilatation, Lipschitz constant, or total energy - do not work. In this paper we give such a modular interpretation, two in fact. We introduce the total expansion $\TE(f)\in [0,\infty]$ of a Lipschitz map $f:M\to N$ between Riemannian manifolds, a notion related to the notion of “$k$-dilatation” developed by Gromov, Guth and others. For volume-preserving Lipschitz maps $f:\Tc_0\to\Tc_1$ between $n$-dimensional, flat, unit-volume tori, we prove that $\TE(f)$ is minimized in the homotopy class of $f$ precisely by the affine maps in that class and takes on these the value $d_{X_n}$. We prove similar results for the Hilbert-Schmidt expansion $\HE(f)$, which is a simple integral over $M$ and has more of an $L^2$ flavor.

We obtain a closed explicit formula for the spectral determinant of flat conical metrics on the bielliptic genus-two locus. The metrics are generated by holomorphic one-forms with two simple zeros. For a symmetric reference metric, an exact Klein-four spectral identity reduces the spectral determinant to scalar determinants on spheres and tori; the singular anomaly formula then yields the general case. The resulting formula involves an elementary binary sextic in the coefficients of the one-form and two explicit hypergeometric areas of four-cone metrics on a quotient sphere. We apply this formula to the separating, one-node nonseparating, and simultaneous two-node degenerations of the curve and obtain complete asymptotics of the spectral determinant in all cases. The one-node degeneration has two distinct metric limits, according to whether the limiting one-form is holomorphic or meromorphic. Comparison of the cylindrical cases with the Bismut–Bost asymptotics determines the corresponding constants explicitly; in the separating case it also evaluates the relative determinant appearing in the Müller–Müller formula. As a by-product, the separating asymptotics evaluate the multiplicative constants left undetermined in earlier general conical and variational determinant formulas.

We study reductions of the tangent bundle of $S^{\dim G}$ through the adjoint representation of a compact connected Lie group $G$. If $d=\dim G$ and $r=\operatorname{rank} G$, we show that the adjoint map $\operatorname{Ad}\colon G\longrightarrow \mathrm{SO}(d)$ is homotopic, as an ordinary map, to one with values in $\mathrm{SO}(d-r+1)$. It follows that every vector bundle over a sphere associated to a principal $G$-bundle via the adjoint representation admits $r-1$ linearly independent sections. Combined with Adams's theorem on vector fields on spheres, this gives the necessary condition $r\le \rho(d+1)$ for an adjoint reduction of $TS^d$. In particular, no such reduction exists for a non-trivial compact connected group, simple or not, of dimension $d>1$ with $d\not\equiv3\pmod4$. As an application, this excludes the adjoint $E_7$-reduction of $TS^{133}$ that is the exceptional branch in the structure-group argument of Bor, Hernández-Lamoneda, Jiménez-Desantiago and Montejano for Banach's isometric subspace problem.