Papers

For integers $1\le k < n$ and real numbers $c_1, \ldots, c_k>0$ and $d_1, \ldots, d_n\ge 0$, we investigate the space of all $k\times n$ matrices $F$ such that the product $FF^*$ is equal to the diagonal matrix ${\rm Diag}(c_1, \ldots, c_k)$ and the squared norms of the columns of $F$ are equal to $d_1, \ldots, d_n$ respectively. Depending on the field where the coefficients of $F$ are taken from, which can be of real or of complex numbers, we are mainly interested in determining whether the resulting space is path-connected or even simply connected relative to the subspace topology in the space of all $k \times n$ matrices. The criteria presented are closely related to results previously obtained by Cahill, Mixon, and Strawn (2017), Needham and Shonkwiler (2021), the last two authors together with Caine (2026), and the author of this work (2024 and 2026). Flag manifolds, namely orbits of the canonical conjugation actions of ${\rm O}(n)$ and ${\rm U}(n)$ on the spaces of symmetric real and Hermitian $n\times n$ matrices, respectively, play a central role in our development.

There exist three nonequivalent left invariant Lorentzian metrics on the Heisenberg group $H_{2n+1}$, or equivalently, three nonequivalent Lorentzian inner products on the Heisenberg Lie algebra $\mathfrak{h}_{2n+1}$, denoted by $\mu$, $\nu$, and $\phi$. We show that, in a specific case, $\mu$ is an algebraic Ricci soliton that is shrinking. Moreover, $\nu$ is an algebraic Ricci soliton only on the three-dimensional Heisenberg Lie algebra $\mathfrak{h}_3$ and it is shrinking. Finally, we show that $\phi$ is a steady algebraic Ricci soliton on $\mathfrak{h}_{2n + 1}$ for $n > 1$. However, for $n = 1$, $\phi$ is flat.

In 1989, Dimitrić showed that any pseudo-umbilical biharmonic submanifold in a Euclidean space is minimal when the dimension of the submanifold is different from four. However, the four-dimensional case has remained open for nearly four decades. This difficulty comes from the fact that the tangential part of the biharmonic equation vanishes identically in dimension four. Therefore, we consider biminimal submanifolds, whose defining equation generalizes the normal part of the biharmonic equation. In this paper, we show that any connected pseudo-umbilical 0-biminimal submanifold of codimension two and dimension at least three in a Riemannian manifold of constant sectional curvature $c$ has constant mean curvature. Furthermore, when $c$ is non-positive, we prove that it is minimal. As a corollary, we show that any four-dimensional pseudo-umbilical biharmonic submanifold in a six-dimensional Riemannian manifold of non-positive constant sectional curvature is minimal. These results also provide partial affirmative answers to Chen's conjecture, generalized Chen's conjecture for non-positive constant sectional curvature, and the Balmuş-Montaldo-Oniciuc conjecture.

We prove the generalized Cartan-Hadamard conjecture, also known as the Aubin conjecture, in the small volume regime under a lower Ricci curvature bound, for any dimension $n\geq 2$. More precisely, we show that if $(M^n,g)$ is a Cartan-Hadamard manifold satisfying $\mathrm{Sec}_g\leq\bar k\leq 0$ and $\mathrm{Ric}_g\geq (n-1)\underline k$, then its isoperimetric profile is at least that of the model space of curvature $\bar k$ for small enough volumes. We also reduce the proof of the generalized Cartan-Hadamard conjecture to establishing rigidity in the equality case.

We study complete stable minimal hypersurfaces in Riemannian manifolds under various curvature assumptions. In an $(n+1)$-dimensional complete oriented manifold with nonnegative scalar curvature, we prove that no complete oriented noncompact stable minimal hypersurface can be conformally equivalent to a bounded domain in an $n$-dimensional manifold with nonpositive scalar curvature. As a consequence, there exists no complete stable minimal hypersurface in $\mathbb{R}^{n+1}$ that is conformally equivalent to a bounded domain in $\mathbb{R}^n$. We also show that compact stable minimal hypersurfaces in manifolds with nonnegative scalar curvature have nonnegative smooth Yamabe invariant and we characterize the equality case. Next, we consider complete two-sided stable minimal hypersurfaces in ambient manifolds with pinched sectional curvature. Under an additional curvature condition that makes the second fundamental form a Codazzi tensor, we obtain a rigidity result under an $L^2$-condition on the second fundamental form and derive an upper bound for the first eigenvalue of the Laplacian. Finally, we establish that any complete noncompact two-sided minimal hypersurface immersed in a warped product manifold is stable, provided that the angle function is positive and the second derivative of the warping function is nonnegative.

We establish a weighted extension of the Bishop-Gromov volume comparison theorem for complete Riemannian manifolds with Ricci curvature bounded below. Given a point $p$ and positive radial weights $f$ and $h$, we consider the weighted volume of a geodesic ball and a corresponding model weighted volume in the simply connected space form of constant sectional curvature $k$. We prove that, under the assumption that $h/f$ is nondecreasing, the ratio of these two weighted volumes is nonincreasing with respect to the radius. We also characterize the equality case, showing that equality forces the weight ratio to be constant and the corresponding geodesic ball to be isometric to the model ball. As applications, we derive weighted Bishop-Gromov-type comparisons, annular comparison inequalities, and weighted volume doubling estimates. We further apply the monotonicity formula to obtain weighted area-volume inequalities and sharp weighted isoperimetric-type comparisons for geodesic balls.

We survey recent developments on the null energy condition (NEC) from the perspective of optimal transport, with emphasis on non-smooth Lorentzian geometry. After recalling the role of classical energy conditions in general relativity, we discuss the relation between Ricci curvature and entropy convexity in optimal transport. The main focus is the authors' optimal-transport characterization of the NEC. In the smooth setting, the NEC is equivalent to displacement concavity of the Shannon entropy power along suitable null-geodesic transport plans supported on null hypersurfaces and measured with respect to rigged measures. We review the geometric ingredients of this characterization and applications, including a weighted Hawking area theorem. We then turn to the synthetic theory introduced in a second paper of the authors. Synthetic null hypersurfaces are defined using achronal boundaries, gauge functions, and reference measures in topological causal spaces. In this framework, the displacement-concavity characterization becomes a synthetic curvature condition, $\mathsf{NC}^{e}(N)$, meaningful without differentiable structure. We discuss compatibility with the smooth theory, invariance under natural changes of gauge and reference measure, and stability under convergence. Finally, we present a synthetic version of Penrose's singularity theorem for continuous Lorentzian metrics. These results indicate that geometric and causal consequences of the NEC persist beyond the smooth setting and highlight optimal transport as a framework connecting curvature, causality, and singularity formation in Lorentzian geometry.

We deduce a new inequality using the Hitchin-Thorpe inequality for noncompact complete four-dimensional manifolds with fibered geometry at infinity and Seiberg-Witten gauge-theoretic techniques. Fibered geometry at infinity is either fibered boundary metric or fibered cusp metric. We prove it for fibered boundary metric when the fiber at boundary is a single point.

By studying the classical Minkowski's inequality, $4\pi A\leq M^{2}$, relating area and mean curvature, in the particular case of convex surfaces of revolution, one is led to a functional inequality that holds for general functions. We state and prove this inequality obtaining a new proof of Minkowski's inequality in this particular case and we establish a sharp inequality between the Minkowski deficit of the surface and the isoperimetric deficit of any meridian curve.

For any volume-collapsing finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface, we show the flow satisfies a Type I curvature bound and classify the corresponding tangent flows. Combined with previous results, this shows that any finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface is of Type I.

Determining the momentum missing on a Poincaré section of a two-degree-of-freedom Hamiltonian is usually posed as the root-finding problem $H(0,q_2,p_1,p_2)=E$, which degenerates at the boundary of the energetically allowed region. We argue that the degeneration is the visible symptom of a geometric object and reorganize the problem around it. The section is a coisotropic hypersurface, the missing momentum is its characteristic direction, and recovering it is a reconstruction through the associated coisotropic reduction. Its carrier is not the fibrewise least-energy function, which we show is informationally incomplete, but the correspondence $\Sigma_E^{\mathrm{sec}}=\Sigma_E\cap C$ together with the reducing projection $\pi_E$, a branched reconstruction over the admissible region whose discriminant is identified with the Hill boundary under a base-side regularity condition. We prove, isolating for each conclusion its minimal hypothesis: that $\pi_E$ is proper and a covering away from the discriminant (coercivity); that the fibres have exactly two points over the interior (unimodality); that along the discriminant $\pi_E$ is a fold (fibrewise nondegeneracy and transversality); and that the discriminant coincides with the topological boundary of the admissible region precisely under a regularity of the least-energy function on the base, which no fibrewise hypothesis supplies. Strict convexity of $H$ in the momenta is shown to be a sufficient but non-fundamental bundle of these conditions. The boundary is a discriminant, not a Lagrangian caustic. Only afterwards does the squared fibrewise energy residual $F_b=(h_b-E)^2$ appear, as the analytic resolution of the fold; inside the admissible region it is a double well with two global minimizers, so branch selection is a separate rule.

A well-known hypersurface classification theorem states that any isoparametric hypersurface in $S^{13}$ with six principal curvatures is homogeneous. This landmark result was first proved in her Annals paper in 2013 by R. Miyaoka with 58 pages, and with errata in Annals in 2016 with 15 pages. The main purpose of this paper is to provide a concise proof of this theorem through an interplay between topological and geometric insights into the Euler class of an oriented vector bundle.

We obtain decay estimates at infinity for Yang-Mills connections and Yang-Mills-Higgs pairs on higher-dimensional manifolds. The main tool is an abstract decay theorem for manifolds with suitable positive harmonic functions. Combining this theorem with refined Kato inequalities, we prove that a natural quadratic decay assumption implies an improved decay rate on certain asymptotically conical and product manifolds.

We study the structure of cyclic Riemannian nilmanifolds, that is, connected nilpotent Lie groups $N$ endowed with a left-invariant cyclic metric $\langle\cdot,\cdot\rangle$. We prove that a Riemannian nilmanifold $(N,\langle\cdot,\cdot\rangle)$ is cyclic if and only if its Lie algebra $\mathfrak{n}$ is at most two-step nilpotent and the associated family of skew-symmetric endomorphisms $J_{\mathfrak{z}}=\{J_Z : Z\in \mathfrak{z}\}\subset so(\mathfrak{a})$ is Abelian. As a direct consequence, every cyclic Riemannian nilmanifold is naturally reductive. We also determine the full isometry group of a connected and simply connected cyclic Riemannian nilmanifold without Euclidean factor.

A long-lasting conjecture in non-Kähler geometry says that any compact Hermitian manifold with constant holomorphic sectional curvature must be either Kähler or Chern flat. The conjecture is known to be true in dimension 2 but remains open in dimensions 3 or higher, except in several special cases. In this article we consider compact Hermitian manifolds that are Bismut torsion-parallel (BTP for brevity), which means that the Bismut connection has parallel torsion. For BTP manifolds that are not balanced, the conjecture was proved by Chen and Zheng. They also confirmed the conjecture for balanced BTP manifolds in dimension 3, utilizing a classification result for such threefolds by Zhao and Zheng. In a recent work, we confirmed the conjecture for balanced BTP manifolds in dimension 4 through a detailed case-by-case analysis. In this article, we establish the conjecture for balanced BTP manifolds in all dimensions. We also disprove the full-rank $B$ conjecture by constructing a compact balanced BTP fivefold with full-rank $B$ that is neither Kähler nor Chern flat; its Chern holomorphic sectional curvature is nonconstant.

A new scheme for parameter pruning is introduced, derived from the differential-geometric distance in model space. Pruning a parameter sets its value to zero, representing a displacement of the model to the hypersurface on which that parameter vanishes. The minimal distance from the unpruned model to this hypersurface is naturally computed via the geodesic distance in the model space as determined by the Fisher information metric. This distance determines the true change in the model, and its performance, under pruning. By analysing progressively more faithful approximations of this geodesic distance a natural hierarchy of optimality for pruning methods is determined. This starts with the traditional magnitude pruning, then develops into new more sophisticated and effective pruning schemes. The method is demonstrated for both fully-connected networks and vision transformers, on MNIST and CIFAR-10, over the complete $0$-$100\%$ pruning range and across five random seeds. It outperforms pruning by parameter magnitude and by the local Fisher information alone in every architecture and dataset combination considered, on both accuracy and the Matthews correlation coefficient. Additionally, analysis of different levels of geodesic approximation produces intermediate pruning schemes that are computationally efficient and maintain near-optimal performance. This geometric picture supplies not only a state-of-the-art pruning methodology for AI models, but also a verified and mathematically-motivated justification for pruning schemes.

On a symplectic manifold, any differential two-form has a natural decomposition into two components: a primitive part and a non-primitive one. Applying this decomposition to the curvature two-form of a principal bundle over a symplectic manifold, we obtain a natural splitting of the Yang-Mills (YM) functional into two functionals that intrinsically depend on the symplectic structure: the primitive Yang-Mills (PYM) functional and the trace Yang-Mills (TYM) functional. We work out the basic properties of the critical solutions of these two functionals. The PYM functional in particular exhibits many of the desirable properties of the YM functional, including the ellipticity of its Euler-Lagrange equations and an algebraic classification of its flat solutions on G-bundles. We also prove a monotonicity formula for the PYM functional as a first step towards characterizing its moduli space of solutions.

We prove Llarull's scalar curvature rigidity theorem for spheres and scalar-mean rigidity for strictly convex Euclidean domains in all dimensions without the spin assumption.

We study an adapted Ricci flow of connections with metric torsion on surfaces with positive Euler characteristic. We first prove that there do not exist any nontrivial solitons of the flow on the $2$-sphere thus confirming a conjecture of Branding–Kröncke (J. Geom. Anal. 27.3 (2017), arXiv:1606.09121). We give an explicit family of torsion data for which the corresponding global solutions fail to converge on $\mathbb{S}^2$. Nevertheless, we provide several sufficient conditions for the convergence of the flow to a stationary point. We first prove that the normalized adapted Ricci flow always converges on $\mathbb{RP}^2$, which completely answers a question in the paper of Branding and Kröncke. Using this, we deduce that the flow converges on $\mathbb{S}^2$ whenever the initial metric and the torsion one-form are antipodally symmetric. We also prove a Łojasiewicz–Simon gradient inequality for the flow and use it to prove convergence to a stationary point provided the solution is close to an arbitrary stationary point.

We solve the long-standing problem of determining the exact denominators of the coefficients of the logarithm $B(x)=x+\sum_{n\geq1}b_nx^{n+1}$, obtained from the universal formal group of complex cobordism by the modulus square construction: $$ b_n=\frac{C_n}{d_n},\qquad d_n=\frac{n+1}{2}\operatorname{lcm}(1,\ldots,2n+2), $$ where $C_n\in\Omega_{\mathrm U}^{-4n}$ is primitive and undecomposable. We prove that $C_n$ belongs to the coefficient ring $\Lambda$ of the universal two-valued law and to the subring $\Lambda_{\mathrm{St}}$ generated by quotient Stong manifolds, and give an explicit integral Stong manifold formula for it. The rational lift of $b_n$ to quaternionic cobordism modulo torsion has exact denominator $2d_n$. Consequently, $C_n$ has no integral quaternionic lift, whereas $2C_n$ does. Every Chern number of $C_n$ is divisible by $d_n$, and $c_{2n}(C_n)=d_n$. We introduce odd genera $\mathrm{Bc}_N$. For $N\geq n$, $\mathrm{Bc}_N$ detects the odd part of $d_n$. Its restriction to the Stong ring is integral exactly for $N\leq3$, while the universal odd genus $\mathrm{Bc}_{\infty}$ takes values there with only odd denominators. For every $n\geq5$, we construct undecomposable classes in $\Lambda^{-4n}$ with equal Ochanine genera and top Chern numbers but distinct $\mathrm{Bc}_3$ values. Finally, the oriented extension of $\mathrm{Bc}_3$ is integral on closed spin manifolds of real dimension below $24$ and on closed string manifolds of real dimension at most $24$. The spin bound is sharp: an Anderson-Brown-Peterson spin $24$-manifold has nonintegral $\mathrm{Bc}_3$ genus.

We construct metrics with $k$-positive Ricci curvature on $\mathbb{T}^k\times \mathbb{S}^m$, for $k,m\geq2$, arbitrarily close to the standard flat-round product in the smooth topology. The two-torus case disproves Wolfson's conjecture that every closed manifold with two-positive Ricci curvature has virtually free fundamental group. We also establish a local obstruction to two-positive Ricci curvature near closed backgrounds carrying three orthonormal parallel one-forms.

We give a Green-function proof of the stable Bernstein theorem in $\mathbb R^7$ for smooth, connected, complete, two-sided minimal hypersurface, thus resolving the last case in stable Bernstein problem.

We prove that every rational proper holomorphic map from $\mathbb{B}^2$ to $\mathbb{B}^4$ has degree at most $5$, confirming D'Angelo's conjecture in this case. This sharp bound follows from Meylan's degree estimates and an intersection-theoretic argument ruling out degree $6$.

Let $(M,\nabla)$ be a real analytic Koszul manifold. An adapted complex structure (ac-structure) on a neighborhood $N$ of the zero section in $TM$ is a complex structure on $N$ such that the leaves of the Levi-Civita foliation are holomorphic curves. More generally, a complex polarization $P$ on $N$ is called an ac-polarization if the leaves of the Levi-Civita foliation are tangential to $P$. The bundle of (1,0) tangent vectors of an ac-structure is an ac-polarization. The connection is called entire (resp. polarization complete or simply $\mathcal P$-complete) if the ac-structure (resp. the ac-polarization) exists on $TM$. Although on a small enough $N$ an ac-structure always exists, entire connections are rear and if a maximal domain of definition $N_{max}$ of an ac-structure exists (different from $TM$), $N_{max}$ is a complicated domain. On the other hand in many cases the associated ac-polarization can be extended to the whole $TM$. The main purpose of the paper is to gain better understanding of this phenomenon using a generalized version of the polar map. As special cases we show that many of those metrics studied by Aslam-Burns-Irvine and Halverscheid-Iannuzzi although are not entire but are $\mathcal P$-complete.

On a connected compact Kähler manifold without boundary, we derive a uniform upper bound for each Laplace eigenvalue over all Kähler metrics in a given Kähler class. This provides an affirmative answer to a question of Yau.

We prove that triangles minimize the Monge-Ampère eigenvalue among bounded planar convex domains of a given area, thus resolving the two-dimensional case of Le's simplex conjecture for the Monge-Ampère eigenvalue (The eigenvalue problem for the Monge-Ampère operator on general bounded convex domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018)). The proof uses a neighbor-decreasing property of the Monge-Ampère eigenvalues of convex polygons along vertex elimination parallel chord movements.

We prove that every closed Lipschitz curve in Euclidean space extends to a Lipschitz disc of least area, with Lipschitz constant controlled by that of the boundary curve. Our proof gives a new approach to the classical Douglas–Radó solution of the Plateau problem but accommodates arbitrary self-intersections and avoids Sobolev maps. The argument extends to metric spaces with curvature bounded above in the sense of Alexandrov.

We show the homology cobordism group of homology $3$-spheres is not generated by graph homology $3$-spheres. Our technique is a combination of filtered instanton Floer theory and Gromov's simplicial volume.

We prove a local stability criterion for the no focal points property on compact Riemannian surfaces. More precisely, if a smooth metric $g$ has non-positive Gaussian curvature and its zero-curvature set is finite, then $g$ belongs to the $C^2$-interior of the set of metrics without focal points. Thus, every sufficiently small $C^2$-perturbation of $g$ still has no focal points, although arbitrarily small perturbations may create regions of positive Gaussian curvature. By Ruggiero's characterization of the $C^2$-interior of the set of metrics without conjugate points, all metrics in the resulting neighborhood are Anosov. We also show that, starting from any hyperbolic metric on a compact surface, one can prescribe an arbitrary finite set as the zero set of the Gaussian curvature of a smooth conformal non-positively curved metric. These metrics can be approximated smoothly by negatively curved metrics, so they lie on the boundary of the negatively curved regime while remaining interior points of the no-focal-points regime. The proof of the stability theorem combines uniform local convexity, Gulliver's bound on the length of geodesic segments contained in small balls, and an inductive argument on the Riccati equation that controls successive passages through the regions where positive curvature may appear.

Let $(M,g)$ be a compact oriented Einstein four-manifold with Einstein constant $E$ and let $\widehat{R}^+$ denote the action of the Riemann curvature tensor on self-dual two-forms. We show that if $\widehat{R}^+ < 0$, then $g$ is strictly linearly stable for the Einstein-Hilbert functional, thus giving a chiral criterion for stability. Our result is stronger than the one by Fine-Krasnov-Singer, who conclude local rigidity from $\widehat{R}^+ < 0$ by proving stability for a different action functional. Our proof proceeds by showing that $(M,g)$ admits a natural spin$^h$ structure carrying a non-zero parallel spin$^h$-spinor. We then apply a lower bound on the Lichnerowicz Laplacian on traceless symmetric two-tensors in the presence of such a spinor.