Papers

Using different approaches, Simon Donaldson and Dashen Yan recently constructed $\mathbb Z_2$-harmonic functions on $\mathbb R^3$ and on $\mathbb R^n$ for $n\geqslant 3$, respectively. Their examples have quadratic growth at infinity and $\mathcal{O}(r^{3/2})$ local growth near a smooth codimension-two branching set $\Sigma$. We show that these properties uniquely characterise such $\mathbb Z_2$-harmonic functions up to rigid motions of $\mathbb R^n$.

Let $S_g$ be a closed oriented surface of genus $g\ge2$. For a reductive representation $\rho:\pi_1(S_g)\to\PU(2,1)$, let $E_\rho$ be the energy function on Teichmüller space associated to equivariant harmonic maps into $\CH^2$. For every positive integer $d$ with $3\nmid d$, all sufficiently large $h$, and every $g>h$, we construct an irreducible reductive representation \[ \rho_{g,h,d}:\pi_1(S_g)\to\PU(2,1) \] with \[ \tau(\rho_{g,h,d})=2h-2-\frac{2d}{3}\notin\mathbb Z, \qquad \operatorname{Crit}(E_{\rho_{g,h,d}})=\varnothing. \] Consequently, the associated branched-minimal-surface forgetful map is not surjective in these nonintegral Toledo components.

Reconstruction problems lie at the very heart of both mathematics and science, posing the fundamental challenge: How does one reconstruct a hidden structure from incomplete, fragmented, or distorted data? In this paper, we introduce a new approach that harnesses the insights of the Vaisman Atiyah–Molino framework. In contrast to conventional methods that depend on persistent homology, our approach exploits the concept of the Vaisman centroid—an intrinsic invariant that encapsulates the averaged geometry of a data set—to resolve the inherent ambiguities of inverse problems. In the present paper, we focus on the theory and applications of the Vaisman centroid, offering a new perspective for Topological Data Analysis that eschews persistent homology in favour of a unified geometric paradigm. A subsequent paper will extend these ideas to a complete reconstruction scheme via the Atiyah–Molino framework. Our method provides a robust and computationally tractable framework for the recovery of hidden structures while opening new avenues for the analysis of high-dimensional and noisy data across the mathematical sciences.

We prove that, on a smooth manifold $M$ equipped with a constant rank horizontal distribution $H$ and a smooth inner product $g_H$ on $H$, any horizontal vector field can be written as a linear combination of, at most, $3N$ Lie brackets of horizontal gradient fields of depth-$(1)$, where $N$ is the minimal immersion dimension of $M$, recovering the Riemannian case trivially when $H$ is the entire tangent bundle. When $H$ is also bracket-generating with a uniformly bounded step, we show that any vector field can be written as a linear combination of a uniformly bounded number of iterated Lie brackets of horizontal gradient fields with uniformly bounded depth, both bounds depending solely on the immersion dimension $N$ and the uniform upper bound on the step of $H$. Utilizing this, in conjunction with the Trotter property, we show that, if the manifold is compact, connected, and without boundary, the group generated by flows of horizontal gradient fields is dense in the identity component of the diffeomorphism group. When the manifold is also Riemannian, we show that the same density statement holds for the group generated by diffeomorphic optimal mass transport maps.

Let $(M,\omega)$ be a compact Hermitian manifold, and let $\Gamma$ be a symmetric convex cone. We develop a quantitative regularization method for $\Gamma$-admissible functions. As an application, we prove Hölder continuity for every pluripotential solution of complex $m$-Hessian equations whose right-hand sides belong to $L^p$, for $p>\frac{n}{m}$. These results extend to a more general class of complex Hessian equations satisfying the structural condition used by Guo–Phong–Tong.

We study metric completions of conformally singular Riemannian metrics on the interior of a compact manifold with corners. Given boundary-defining functions \(\rho_1,\ldots,\rho_N\) and nonnegative weights \(\alpha_1,\ldots,\alpha_N\), we compare the product metric \(g_\Pi=(\prod_i\rho_i^{-\alpha_i})g_0\) with the sum metric \(g_\Sigma=(\sum_i\rho_i^{-\alpha_i})g_0\). For a boundary point \(p\), let \(I(p)\) be its active index set, \(A(p)=\sum_{i\in I(p)}\alpha_i\), and \(M(p)=\max_{i\in I(p)}\alpha_i\). We prove that \(p\) lies at finite metric distance precisely when \(A(p)<2\) for the product metric and \(M(p)<2\) for the sum metric. Over every accessible boundary point, the active normal directions collapse to a single completion point, yielding a canonical topological description of the completion. The interaction laws produce different boundary-incidence behavior: product weights can make a deeper face inaccessible even when its constituent hypersurfaces are individually accessible, whereas the sum model is governed by the largest active weight. We also determine the induced geometry on accessible open faces: locally the boundary metric is a snowflake with exponent \(1-A_I/2\) in the product case and \(1-M_I/2\) in the sum case, giving Hausdorff dimensions \(m/(1-A_I/2)\) and \(m/(1-M_I/2)\), respectively.

We study the classification of scalar generalized bi-Hamiltonian structures with semisimple leading terms. As an application, we prove the existence of integrable viscous conservation laws for any given viscous central invariant.

We establish optimal Casorati inequalities for pointwise slant Riemannian submersions. As such a submersion carries two distinct distributions, the vertical, tangent to the fibers, and the horizontal, governed by O'Neill's tensors T and A, we treat them separately. For submersions from generalized complex and generalized Sasakian space forms we bound the normalized scalar curvature of each distribution by its normalized Casorati curvatures. Special cases recover sharp inequalities for real, complex, Kähler, Sasakian, Kenmotsu, cosymplectic, and almost and almost space forms. Equality is characterized geometrically: invariantly quasi-umbilical fibers in the vertical case and integrability in the horizontal case. The invariant and anti-invariant limits reproduce known results, and examples illustrate sharpness.

We determine the Gromov–Hausdorff compactification of the moduli space of unit-diameter Ricci-flat Kähler metrics on Enriques surfaces. We show that all metric degenerations are encoded by the adjoint Satake compactification of the Enriques period domain. There are 32 boundary components added in this Satake compactification, and the GH limit at each boundary is determined. As a corollary, all the GH limits of Kähler Ricci-flat metrics on Enriques surfaces are classified, including those for a fixed complex structure or a fixed polarization.

We construct complete, one-ended Riemannian manifolds in dimensions four and five having strictly positive Ricci curvature, with fundamental group isomorphic to the Prüfer $2$-group $C_{2^\infty}$ and any subgroup $\Gamma\leq \mathbb Q/\mathbb Z$, respectively. This gives counterexamples to Milnor's conjecture in the two remaining dimensions.

We construct smooth toric Fano $n$-folds of Picard number $n+2$ whose tangent bundles are slope-stable with respect to the anticanonical polarization, for every $n\ge4$. The construction gives toric fibrations with fibre the del Pezzo surface of degree six over products of projective lines, parametrized by multisets of roots of $A_2$. Every nonempty multiset of nonzero roots with vanishing sum yields a stable tangent bundle. Among these multisets, the resulting variety is Kähler-Einstein if and only if the multiset is invariant under negation or under the order-three rotation of the root hexagon. We also construct stable, non-Kähler-Einstein examples with nonzero twist sum in every even dimension at least four. The proofs reduce slope inequalities and barycenter computations to integrals over the moment hexagon. For a smooth toric Fano variety with strictly semistable tangent bundle, we prove that polystability is equivalent to decomposition as a nontrivial product of smooth toric Fano varieties with anticanonically stable tangent bundles. Exact evaluation of Klyachko's criterion extends the stability classification to dimensions five and six, recovering the classifications of Steffens and Reynolds in dimensions three and four. Together with the product criterion, this determines polystability for all $8{,}630$ varieties in dimensions three through six, and hence the existence of Hermitian-Einstein metrics on their tangent bundles with respect to an anticanonical Kähler form.

We characterize all symmetric, embedded surfaces of revolution with constant mean curvature that meet the unit sphere orthogonally. It is well known that there exists a unique free boundary catenoid inside of the unit ball. By analogy, we prove the existence of a unique free boundary nodoid inside of the unit ball with constant mean curvature equal to $-1$. Moreover, there exists a unique compact free boundary nodoid outside of the unit ball with constant mean curvature equal to $-1$. However, there exists no symmetric surface of revolution with constant mean curvature $1$ that meets the unit sphere orthogonally.

In this article, we consider discontinuous martingales on tangent bundles over submanifolds of Euclidean space. First, we introduce a connection rule on tangent bundles and establish the Itô calculus for discontinuous semimartingales on tangent bundles. Then we focus on harmonic maps with respect to non-local Dirichlet forms and show that the derivative of harmonic maps along infinitesimal symmetries induces discontinuous martingales on tangent bundles. This process may be viewed as a stochastic Jacobi field along the image martingale. We also introduce the stochastic parallel transport of tangent vectors along càdlàg semimartingales on Riemannian submanifolds with projected jumps. Using the parallel transport, we obtain the mean-value property for the differential of harmonic maps involving a jump part expressed through the second fundamental form. We also obtain a derivative formula for harmonic maps for isotropic Lévy processes with a Brownian component on compact Riemannian manifolds.

We show that the stable norm of flat metrics on $H_{d}(T^n,\mathbb{R})$ is locally rigid if $1\leq d<n-1$ and locally rigid among metrics of the same volume if $d=n-1$. We also show that the stable norm on $H_{2}(T^3,\mathbb{R})$ is not locally rigid. As applications, we answer negatively a question raised by Bangert in his ICM address, prove local rigidity of the marked $k$-area spectrum of flat metrics for $1\leq k\leq n-2$, and prove a local rigidity result for the volume spectrum of flat metrics on $T^n$.

In this paper we extend Hitchin-Kobayashi correspondence along holomorphic fibre space and we show that there exists a generalized Hermitian-Einstein metric which is twisted with Weil-Petersson current.In fact we generalise Song-Tian program for vector bundles

The covariance matrix of a basket of assets is a symmetric positive-definite object that evolves on a curved manifold, not on a flat vector space. Standard linear regression on its vectorised entries ignores that geometry and, in periods of market stress, can return fits that fail to be positive definite. We study geodesic regression on the symmetric positive-definite cone $\SPD(n)$ under the log-Euclidean and affine-invariant metrics, with the rolling sample covariance of a basket of log-returns as data. The fitted geodesic's velocity norm is proposed as a geometry-aware turbulence indicator. On daily data for four equity baskets- a ten-stock historical S&P sub-basket (2006-2024), a current S&P~100 mega-cap basket, the top ten listings of the Johannesburg Stock Exchange (JSE), and the ten most-traded Egyptian Exchange (EGX) tickers the indicator peaks cleanly on known stress episodes (the 2008 global financial crisis, the 2020 COVID crash, the 2022 Fed rate-hike cycle, and the October-November 2019 Egyptian political-economic tension episode) without any calibration. On hold-out evaluation in crash windows, log-Euclidean regression achieves 150 to 300 times smaller geodesic mean squared error than Euclidean ordinary least squares, because the latter produces non-SPD forecasts that blow up under the intrinsic metric. A naive velocity-aware de-risking strategy reduces maximum drawdown by 2.5 percentage points at the cost of 1.4 percentage points of annual return, consistent with a slow-moving indicator that signals stress reliably but translates imperfectly into tactical trading. All data, code, and experiments are reproducible from the companion repository.

We present a unified construction of all simply connected irreducible exceptional compact symmetric spaces as totally geodesic submanifolds of $E_{8}$. Using this construction, we obtain a unified description of totally geodesic embeddings between exceptional symmetric spaces by determining all inclusion relations among them. The resulting inclusion structure exhibits a remarkable and unexpected symmetry, suggesting a deeper geometric relationship among exceptional symmetric spaces.

We investigate integrability of Euler-Lagrange equations associated with 3D second-order Lagrangians of the form \begin{equation*} \int f(u_{xy},u_{xt},u_{yt})\ \text{d}x\text{d}y\text{d}t. \end{equation*} It is demonstrated that there are exactly four different types of such Lagrangian densities $f$: the first one is given by the formula $f=\sqrt{u_{xy}u_{xt}u_{yt}}$, the second and the third are more complicated (although still representable in elementary functions), whereas the most generic fourth one is expressible in terms of the Lobachevsky function, revealing unexpected links to spherical/hyperbolic trigonometry and Schläfly-type formulas. Dispersionless Lax pairs and integrable dispersive deformations of the corresponding Euler-Lagrange equations are also constructed.Remarkably, dispersive deformation of the first Lagrangian density coincides with the Lagrangian of the classical Darboux system arising in the theory of triply-orthogonal coordinate systems in $\mathbb{R}^3$. Dispersive deformations of the three other cases provide Lagrangian formulation of semi-discrete and fully discrete versions of the Darboux system, with one, two and three discrete variables, respectively.

We prove a sharp $2$-systolic inequality for four-dimensional products $\mathbb S^2\times P$, where $P\subset\mathbb R^2$ is an arbitrary convex polygon. Let $h=g_{\mathbb S^2}+g_{\mathrm{eu}}$ be the standard product metric. If a Riemannian metric $g$ on $\mathbb S^2\times P$ has scalar curvature $\geq \sigma>0$, nonnegative mean curvature on every codimension one face, and dihedral angles no larger than the corresponding dihedral angles of $h$, then both its homotopy and homology $2$-systoles are at most $ \frac{8\pi}{\sigma}.$ This confirms a conjecture of Gromov.

In the previous paper \cite{Jin26}, the author introduced the notions of an adapted current $T$ and an adapted Hermitian-Einstein metric to establish the Kobayashi-Hitchin correspondence for a nef and big class $\alpha$. As a continuation of the previous work, this paper studies the solvability and the convergence of the Yang-Mills flow for a nef and big class $\alpha$ on a holomorphic vector bundle $E$ over a compact Kähler manifold $X$. In particular, we show that the limit of the Yang-Mills flow at infinity is determined by the holomorphic structure of $E$ and the nef and big class $\alpha$. More precisely, if we fix an integrable unitary connection $A_0$ on $E$, we show that the $T$-Yang-Mills flow on $E$ with initial condition $A_0$ is solvable for all time and it converges to a $T$-Yang-Mills connection $A_{\infty}$ in the sense of Uhlenbeck limit. Furthermore, we also show that, on the ample locus of $\alpha$, $A_{\infty}$ is complex-gauge equivalent to the direct sum of the Chern connections of the $T$-adapted Hermitian-Einstein metrics on the factors of the graded sheaf associated with the $\alpha^{n-1}$-Harder-Narasimhan-Seshadri filtration of $E$.

An admissible configuration $\Lambda\subset\Hh^m\simeq\Rr^{4m}$ defines a compact intersection of quadrics $Z_{\Hh}(\Lambda)$ and a free diagonal $\Sp(1)$ quotient $N_{\Hh}(\Lambda)$. We realize this quotient as the leaf space of a real dilation action on an open subset of quaternionic projective space. The total space is a quaternionic moment-angle polyhedral product. Every quotient is $2$-connected; when there are no indispensable coordinates, it is $3$-connected and its fourth integral cohomology is generated by the Euler class $e$ of the associated oriented rank-four bundle. We compute $p_1(TN_{\Hh})=2(n-2)e$ and prove that no LVMQ manifold admits a quaternionic Kähler metric. An explicit infinite family consists of products of spheres admitting both homogeneous positive Einstein metrics and quaternionic toric structures in the sense of Gentili–Gori–Sarfatti. We also describe the trace foliations of the real action in the Poincaré domain. For the separate rank-one distribution $X\Hh$, an invertible quaternionic-linear field is integrable only in the real scalar case. A nonlinear field with positive real scalar derivative at the origin and involutive $X\Hh$ has exactly the standard quaternionic Hopf foliation on every sufficiently small centered sphere.

Neurofinance shows that financial valuation depends on evolving neural states, while temporal experience is itself state dependent. Yet intertemporal models typically treat time as exogenous and ask how delay affects valuation. This paper examines the converse question: can valuation-related neural dynamics generate subjective financial time? We develop a continuous-time model on a joint financial–neuro-evaluative state space in which subjective financial time is accumulated valuation along dynamically admissible histories. We characterise realised temporal-rate dispersion and show that, under matched financial dynamics, distinct neuro-evaluative trajectories can generate different subjective financial times. Financial equivalence therefore need not imply temporal equivalence. Valuation curvature further determines local path dependence. A proof-of-concept behavioural–fMRI analysis uses \(1{,}183\) observations, \(1{,}126\) valuation transitions, and \(798\) financially matched pairs. Using a medial-prefrontal valuation-state coordinate based on vmPFC and dmPFC responses, the strongest participant exhibits a positive association between neural-state separation and subsequent valuation divergence (\(\rho=0.253,\ p_{\mathrm{perm}}=0.001\)), robust to residual financial distance (\(\rho=0.251\)) and tighter matching (\(\rho=0.207\)). Participant effects are heterogeneous, although combined evidence gives \(p=0.0216\). The data support the neural state-separation premise rather than directly identifying subjective financial time. The framework establishes a theoretical and empirically grounded basis for endogenous subjective time in neurofinance. \keywords{Neurofinance $\cdot$ Subjective financial time $\cdot$ Neural valuation $\cdot$ Relative subjective value $\cdot$ Intertemporal choice $\cdot$ Temporal dispersion $\cdot$ Path dependence

These notes are devoted to the kinetic Fokker–Planck equation on the tangent bundle of a compact Riemannian manifold, with motivations coming from relativistic kinetic theory of gases (à la Debbasch) and geometric analysis (à la Bismut). They are developed in particular from collaboration with Fabrice Debbasch and Yann Ollivier. The presentation is mostly self-contained, intended for readers without much prior familiarity with geometric analysis. Once the kinetic equations are reviewed, I expand the basic toolbox from functional and spectral analysis. Then the main focus is on global hypoelliptic $C^\infty$ regularization and hypocoercive exponential equilibration, both of which are addressed both in $L^2$ and $L^1$ settings. Various types of interpolation inequalities, spectral gap and entropic inequalities are established and used. Several existing results are simplified and unified, along with new ones. Some directions of research are mentioned.

Existence of canonical metric on a Sasakian variety was a long standing conjecture and the major part of this conjecture is about varieties which do not have definite basic first Chern class . Most of the Sasakian varieties do not have definite basic first Chern class. We give a program by using Fujino's Minimal Model program to find canonical metric on the canonical model of Sasakian varieties.

Let $n,m$ be integers such that $n\geq2$ and $0\leq m\leq n-2$. Let $C_{m+1}$ denote the $(m+1)$-intermediate curvature introduced by Brendle–Hirsch–Johne. We prove that there are constants $\nu(n,m),C(n,m)>0$ such that the following holds. If $(M^n,g)$ is complete and connected and, for $\delta\geq 0$, \[ \mathrm{Ric}\geq-\delta^2, \qquad C_{m+1}\geq 1, \] then \[ \delta R\leq\nu(n,m) \quad\Longrightarrow\quad \mathrm{Vol} B_R(p) \leq C(n,m)R^m \quad \text{for every $p\in M$ and $R>0$.} \] In particular, taking $m=n-2$ and $\delta=0$ gives Gromov's conjectured codimension-two volume growth estimate under $\mathrm{Ric} \geq0$ and $\mathrm{Scal} \geq1$.

We introduce and study a Fisher information metric \(g^F_\tau\) associated to the conjugate heat kernel of a Ricci flow \((M^n,g_t)\). This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that \(g^F_\tau\) is monotone in scale and satisfies \(g^F_\tau\le g_t\). We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect \(g_t-g^F_\tau.\) This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities \(0<g^F_\tau<g_t\) at every positive scale, while in the complete case equality forces a Euclidean splitting. We also develop several consequences of this point of view. These include a sharp reverse Poincaré inequality for the heat semigroup, a contraction formula for \(\varphi\)-divergences along conjugate heat flow, and a canonical construction of heat-kernel splitting maps from large eigenvalues of the Fisher endomorphism. As applications, we relate pointed Nash entropy close to \(0\) to small Fisher deficit at comparable scales, and we obtain a codimension-one Fisher-metric criterion for applying Bamler's \(\varepsilon\)-regularity theorem.

We introduce the almost isoclinic region in the oriented Lagrangian Grassmannian ${\rm Lag}^+(n)$ of $\mathbb{C}^n$, an intrinsic higher-dimensional analog of a natural convex region in ${\rm Lag}^+(2) \simeq \mathbb S^1\times \mathbb S^2$. A Lagrangian submanifold is called almost isoclinic if its Gauss map takes values in this region, extending the graphical condition that the characteristic angles of the tangent plane remain uniformly close. We construct a canonical positive function $\Lambda$ on this region and prove that $\log \Lambda $ is concave with respect to the invariant Grassmannian metric. This property yields subharmonicity and monotonicity formulas for minimal Lagrangians and Lagrangian mean curvature flow. As applications, we prove rigidity and Bernstein-type results, including that a complete connected almost isoclinic minimal Lagrangian with a positive lower bound for $\Lambda$ must be a Lagrangian $n$-plane.

Given a pair of (round) 2-dimensional spheres, one of which has radius three times that of the other, the Lie algebra of local infinitesimal symmetries for the distribution corresponding to rolling the spheres along each other (without letting them slip or twist) happens to be isomorphic to the split real form of the exceptional simple Lie algebra of rank 2. These exceptional local symmetries appear only for this specific 1:3 ratio of radii, however, and while there are several proofs of this result, a straightforward kinematic explanation for the seemingly miraculous appearance of an exceptional simple Lie group in this situation has long been desired. In this paper, we provide such an explanation, relating the ratio of radii to the intersections of a pair of one-parameter subgroups that can be seen from the rolling sphere perspective. The approach does not require the split-octonions, as we construct the exceptional simple Lie group directly by Tanaka prolongation to aid in visualizing the underlying geometry.

For a complete Riemannian manifold with nonnegative Ricci curvature, we prove two sharp volume growth order estimates, thereby resolve a conjecture of Gromov in 1986. There first is that a uniform deficit in the volume of unit balls, an analog of positive macroscopic scalar curvature, forces codimension one volume growth, and the second one is that a uniformly positive scalar curvature lower bound forces codimension two growth known as the codimension two volume growth conjecture.

We classify smooth compact connected Lagrangian immersions $X$ in the closed unit ball of $\C^n$, $n\ge2$, satisfying $H+\varepsilon X^\perp=0$, $\varepsilon\in\{-1,0,1\}$, with Legendrian boundary on the unit sphere and constant contact angle on each connected component. We prove that the boundary has at most two connected components. When the boundary is connected, $X$ is a diffeomorphism onto an equatorial Lagrangian $n$-disk. When the boundary has two components, $X$ splits globally as $X(s,p)=\gamma(s)\psi(p)$, where $\psi$ is a compact minimal Legendrian immersion in the unit sphere and $\gamma$ is an Anciaux profile with a unique radial minimum. The two contact angles are supplementary. In complex dimension two, every non-disk solution is a finite cover of a Lagrangian catenoid segment for $\varepsilon=0$ or of a rotational Anciaux annulus for $\varepsilon=\pm1$. In higher complex dimensions, iterated Calabi suspensions produce families whose minimal Legendrian links have nontrivial topology.