Papers

Suppose $(M, g)$ is a smooth, closed, $5$-dimensional Riemannian manifold with positive Yamabe invariant $Y(M, [g]) > 0$ and positive Yamabe-type $Q$-curvature invariant $Y_{4}^{\ast}(M, [g]) > 0$. Using the continuity method, we prove the existence of a metric in the conformal class $[g]$ with positive scalar curvature and positive $Q$-curvature, assuming an additional condition on an “initial metric" in the class $[g]$.

In this paper, we provide new explicit examples of semitoric systems. More precisely, for any natural number $n$, we give a semtioric moment map $F_n$ with $2n$ focus-focus singularities defined on a symplectic manifold $(M_n,\omega_n)$ of diffeomorphism type $\mathbb{CP}^2\sharp(2n+1)\overline{\mathbb{CP}}{}^2$.

We establish flow and homotopy tools for Banach Lie algebroids that do not rely on compactly supported extensions. Under suitable hypotheses, we construct parameter-dependent local extensions of sections along algebroid paths, complete lifts and their fibrewise-linear evolutions, and a Bochner-integral variation-of-constants formula. We then prove that an admissible-path homotopy is equivalently a Lie algebroid morphism from the parameter square and a solution of the associated transport equation. This provides the extension, transport and infinitesimal variation theory required for a local integration program of Banach Lie algebroids to be developed in a sequel.

Retractions provide a computationally efficient alternative to the Riemannian exponential and logarithm maps for practical data-processing tasks on manifolds. In particular, second-order retractions with closed-form inverse are well-suited for interpolation problems on manifolds. On the Stiefel manifold of orthogonal frames, there are only two retractions of this type: the Cayley retraction, which is second-order accurate under the canonical metric, and the recently proposed polar-light retraction, which is second-order accurate under the Euclidean metric. In this paper, we study the properties of these maps in the context of interpolation on the Stiefel manifold. To obtain explicit interpolation error bounds, we examine the conditioning of the retraction maps and their inverses. We show that the retractions are well-conditioned, and we derive interpolation error bounds similar to those of classical Euclidean interpolation. The inverse retractions are not well-conditioned in general, and we discuss how data can be mapped via an isometric group action to ensure stable computations. As with all retractions on compact manifolds, the inverse canonical Cayley retraction and the inverse polar-light retraction exist only locally, and we construct normal neighborhoods around any point in which either the inverse Cayley retraction or the invese polar-light retraction are guaranteed to be computable. As an application of the retraction maps, we consider Hermite interpolation, where the objective is to reproduce both sampled function values and derivative information. A numerical example demonstrates that retraction-based interpolation is competitive with classical methods based on Riemannian normal coordinates.

We introduce the affine reflection transform (ART) of an oval $C$ relative to an interior point $p$ by applying the parallel-tangent involution to chords through $p$ and taking the envelope. A dual-projective approach gives a coordinate-free singularity criterion and proves that every generic non-degenerate ART has an odd number of ordinary cusps, at least three. Its relation to the centre symmetry set and the Wigner caustic yields further cusp bounds. We define a discrete ART for convex polygons with parallel opposite sides. Its combinatorial cells admit explicit rational parametrisations. Non-degenerate cells are arcs of ellipses or hyperbolas, and never parabolas. Under refining tangent approximations, the discrete transform converges in the Hausdorff metric to the smooth one. The oriented area of the smooth and polygonal transforms is non-positive and admits a Sobolev-type interpretation. Maximising its absolute value produces an affine-invariant asymmetry measure and a set-valued affine centre. We establish boundary degeneration and vanishing results, characterise central symmetry, and show that the maximising locus need not lie on the centre symmetry set or Wigner caustic. Finally, we prove universal upper bounds for the normalised energy, strengthen them for ovals of constant width.

Motivated by the geometric interpretation of spatially homogeneous cosmological models, we formulate a spacetime closedness theorem directly at the Lorentzian level. A classical space-form classification theorem organizes homogeneous and isotropic spatial geometries, but by itself it does not yield a Lorentzian statement about the ambient spacetime. The main technical step is to derive finite slice-volume from genuinely Lorentzian control hypotheses on the foliation. This is achieved in a strong version, for maximally regular globally hyperbolic $(n+1)$–spacetimes with a finite-time Big Bang and homogeneous complete spacelike slices, and in a weaker version in which maximal regularity is replaced by time-integrability of the accumulated expansion rate. In both cases, the argument separates an analytic step, deriving finite slice-volume from the Big Bang and temporal control, from a geometric step, upgrading finite volume to compactness by homogeneity and completeness. The theorem is stated in arbitrary spacetime dimension and is accompanied by a Lean~4 formalization of the strong and weak abstract statements.

Sasakian manifolds, the odd-dimensional analogues of Kähler manifolds, carry two natural pairs of first-order Dolbeault-type operators on the full complex of differential forms, extending the Kohn–Rossi differentials. On such Sasaki manifolds, we establish Kähler-type identities for these operators, and relate the resulting Dolbeault Laplacians to the Hodge Laplacian. Unlike the Kähler case, where $\Delta=2\Delta_{\overline\partial}$, the relation has extra terms coming from the Reeb flow. Using these formulas and a Lefschetz decomposition, we derive lower and upper eigenvalue estimates for $\Delta$ on forms depending on the eigenvalues of the Lie derivative of the Reeb vector field.

In this paper, we develop the weighted level set method for a Kähler manifold $(X^n,\omega)$ admitting an almost holomorphic map to a possibly singular base $Z$, which is not uniruled. As a key intermediate result, we prove that any blowup $\operatorname{Bl}_SX$ of $X$ along smooth submanifolds $S$ of $ \operatorname{codim} S\ge2$ admits a sequence of Kähler metrics with scalar curvature globally and arbitrarily $C^0$-close to the scalar curvature of $\omega$. As a consequence, we establish the sharp \(2\)-systole estimate for every positive scalar curvature Kähler manifold $(X,\omega)$ and prove $\min_XS(\omega) \cdot\operatorname{sys}_2(\omega) \le 2\pi r(r+1)$, where \(r\) is the rational dimension of $X$, with equality if and only if the universal cover splits as $(\widetilde X,\widetilde \omega) \cong (\mathbb P^r,\omega_{\mathrm{FS}}) \times(Y^{n-r},\omega_{\mathrm{RF}})$ up to normalization where $\omega_{\mathrm{FS}}$ is the Fubini-Study metric and $\omega_{\mathrm{RF}}$ is Ricci-flat. We also show a sharp even-systolic inequality in the same setting when the general fibre is the projective space.

Let $\mathbf{C}=\partial E\subset \mathbb{R}^{n+1}$ be a regular area-minimizing hypercone. We first prove that $\mathbf{C}$ is simultaneously strictly stable and strictly minimizing if and only if there exists $c_\mathbf{C}>0$ such that \[ \operatorname{Per}(F;B_R)-\operatorname{Per}(E;B_R) \geq c_\mathbf{C} \int_{F\mathbin\triangle E} \frac{\operatorname{dist}(x,\mathbf{C})}{|x|^2}\,dx \] for every $R>0$ and finite-perimeter set $F$ with $F\mathbin\triangle E\Subset B_R$. Thus this intrinsic distance-weighted inequality characterizes exactly the simultaneous strictness of the stability and minimizing properties. Second, without either strictness assumption, every regular area-minimizing hypercone satisfies the scale-invariant quadratic inequality \[ \frac{\operatorname{Per}(F;B_R)-\operatorname{Per}(E;B_R)}{R^n} \geq c_\mathbf{C} \left( \frac{|F\mathbin\triangle E|}{R^{n+1}} \right)^2. \] This extends the inequality previously established for area-minimizing Lawson cones. Finally, the area-minimizing assumption is unnecessary for our spectral result: for every stable regular minimal hypercone, the first Dirichlet eigenvalue $\lambda_\mathbf{C}^D(R)$ of the Jacobi operator on $\mathbf{C}\cap B_R$ is given exactly by \[ \lambda_\mathbf{C}^D(R) = \frac{j_{b_1,1}^2}{R^2}, \qquad b_1^2 = \frac{(n-2)^2}{4}+\mu_1, \] where $\mu_1$ is the first eigenvalue of the link Jacobi operator and $j_{b_1,1}$ is the first positive zero of the Bessel function $J_{b_1}$. In particular, this identifies the optimal Dirichlet spectral constant for every stable regular minimal hypercone.

We prove a sharp partial regularity result for Hamiltonian stationary Lagrangian Lipschitz submanifolds in arbitrary smooth almost Kähler manifolds: every weak solution of the corresponding equation is smooth away from a relatively closed singular set of Hausdorff dimension at most $n-5$. We show that the estimate is optimal by constructing a nonzero two-homogeneous viscosity solution \[ U\in C^{1,1}(\mathbb{R}^5)\setminus C^2(\mathbb{R}^5) \] of the phase-zero special Lagrangian equation, whose level sets on $\mathbb{S}^4$ are the leaves of Cartan's isoparametric foliation. Its gradient graph is a non-flat calibrated cone, real analytic away from the vertex. This also gives the first $C^{1,1}$ but non-$C^2$ solution of the special Lagrangian equation, and shows that the same dimensional estimate is sharp in the case of special Lagrangian graphs.

We show that if $G$ is a real semisimple Lie group and $\Gamma<G$ is a discrete subgroup with slow growth, in the sense that its growth indicator function is smaller than $\rho$, then $\Gamma$ acts totally dissipatively on the Furstenberg boundary of $G$. Moreover, for $G= SO(n,2)$ and $n\geq 3$, we construct infinite-covolume discrete subgroups that act ergodically on the Furstenberg boundary of $G$, providing a counterexample to a conjecture of Margulis for $G = SO(n,2), n > 2$. The examples arise from lattices $\Gamma<H= SO(n,1)$ and their deformations in $G$. Perhaps more importantly, we describe a new approach to studying deformations of such lattices by relating them to deformations of the smooth right-translation action of $SO(n-1,1)$ on $\Gamma\backslash H$. This correspondence allows us to apply recent results of DeWitt and Dolgopyat on smooth group actions and to give examples where the right-translation action of $SO(n-1,1)$ on $\Gamma\backslash H$ can fail to be $C^0$-locally rigid.

We study entire solutions of the Allen–Cahn equation $\Delta_g u = W'(u)$ on Cartan–Hadamard manifolds of the form $u = U\circ b$, where $b$ is a Busemann function and $U\in C^2(\mathbb{R})$; we call these Busemann profiles. Our main result is a rigidity principle requiring no homogeneity of the manifold: if $\inf \Delta_g b>0$ and $U$ has finite limits $\ell_\pm$, then $W(\ell_-)\le W(\ell_+)$, with equality exactly when $U$ is constant. In particular no nonconstant Busemann layer joins the two wells of a balanced double-well potential, in any dimension $n\ge 2$; under $K\le-\kappa_2<0$ the hypothesis is automatic by Hessian comparison, and if $W'$ is locally Lipschitz it holds under the mere nonvanishing of $\Delta_g b$. The mechanism is a dissipation identity for the Hamiltonian $\tfrac12(U')^2-W(U)$ along the flow lines of $\nabla b$, in which the mean curvature of the horospheres acts as a wave speed. Conversely, when $\Delta_g b\equiv c$ and $W$ is bistable and unbalanced, strictly increasing Busemann fronts joining the wells exist precisely when $c$ is the Fife–McLeod speed; their interfaces are then horospheres of constant mean curvature. On warped-line and rotational surfaces we test sharpness: monotone warpings admit no equal-level profile and rotational surfaces no radial two-well profile, while an even convex warping carries a nonconstant monotone layer whose nodal set is the central geodesic leaf.

We show that purely topological torus symmetries impose strong restrictions on the fundamental groups of almost non-negatively curved closed manifolds and on their actions on higher homotopy groups, even when the torus action is unrelated to the metric.

We prove an inverse optimal-transport principle for the ABP contact set of animmersed submanifold. A fixed function on the submanifold determines, for every probability density on a bounded convex target, a source measure and an optimal plan. For a uniform ellipsoidal target, disintegration over the submanifold gives conditional measures whose fiberwise $L^\infty$-norms satisfy a sharp weighted average estimate with constant $\omega_n/\omega_{n+m}$ in codimensions $1\leq m\leq4$. This yields the sharp ellipsoidal Michael–Simon–Sobolev inequality in the same range, including its equality cases and symmetrization consequences. To the best of our knowledge, even in the Euclidean case, the sharp constants in codimensions three and four were previously unknown.

We prove that the notion of mass recently introduced by Mazurowski and Yao for continuous metrics coincides with Huisken's isoperimetric mass on smooth Riemannian $3$-manifolds with nonnegative scalar curvature whose metric has sharp $C^0$-asymptotically flat behavior. As a consequence, we deduce a Riemannian Penrose inequality for the mass parameter of asymptotically Schwarzschildian $3$-manifolds.

We show that the spacelike solutions to the curvature flow for curves on the De Sitter space are in correspondence with the solutions to the inverse curvature flow on the 2-dimensional hyperbolic space, that on the De Sitter space, the timelike solutions to the curvature flow are in correspondence with the timelike solutions to the inverse curvature flow, and that the solutions curve shortening flow on the 2-dimensional hyperbolic space are in correspondence with the solutions to the spacelike solutions to the inverse curvature flow on the De Sitter space. We prove that, for spacelike curves on the De Sitter space, the curvature flow is a gradient-type flow for the arc-length functional. We observe that a spacelike or timelike curve on the De Sitter space is a soliton solution to the curvature flow (resp. inverse curvature flow) if and only if its curvature (resp. inverse of its curvature) can be written as the inner product between its tangent vector field and a fixed vector $v$ of the 3-dimensional Minkowski space. We prove that for each vector $v$, there exists a 2-parameter family of timelike (spacelike) soliton solutions to the curvature flow and to the inverse curvature flow on the De Sitter space. We show that there exists no non-trivial complete timelike soliton. There exist non-trivial complete spacelike solutions. As a consequence of curvature flow, we obtain the behavior of the soliton solutions to the inverse curvature flow on the De Sitter space and hyperbolic space.

It is not a priori clear which of the Gauduchon connections is better suited to LCK and Vaisman manifolds. We thus investigate the geometry of these connections through Einstein problems and more general analytic and cohomological conditions on their Ricci tensors. Our results single out the Bismut connection as the privileged one for the non-Kähler lcK and Vaisman geometry. We therefore study the {second-Bismut–Einstein} equation and provide a characterization of these structures on Hopf manifolds.

We develop a differential graded model for finite non-singular semisimplicial sets and their cellular sheaves. To a finite non-singular semisimplicial set \(S\), we associate an exterior DGA \(\Omega^\bullet_S\), extending the classical anti-equivalence between finite simple directed graphs and first-order differential calculi to higher degrees. We characterize the essential image of this construction, obtaining an equivalence of categories that recovers the graph-FODC correspondence in dimension one. For a fixed \(S\), we then characterize a category of differential graded \(\Omega^\bullet_S\)-modules equivalent to the category of cellular sheaves on $S$, thereby giving a differential refinement of the usual incidence-algebra description. Finally, we study connections and curvature in this framework. Connections are described by edgewise linear maps and their curvature on a 2-dimensional simplex is the difference between direct and composite edge transport. When the edge transports are invertible, vanishing of this curvature on every \(2\)-simplex can equivalently be seen as a gluing criterion to extend the given edge transports to a connection sheaf on \(P_S\).

We study properties of the transcendental numerical dimension on compact Kähler manifolds. In particular, we establish a connection between this invariant and degenerate divisors in fibrations, extending few results from the projective setting. We also study fibrations between compact Kähler manifolds with rationally connected general fibre. We prove that every pseudo-effective line bundle contained in a tensor power of the cotangent bundle comes from a pseudo-effective line bundle contained in the corresponding tensor power on the base, up to an explicit relation involving degenerate divisors. Finally, combining these results with a theorem of Cao–Păun, we answer a question they posed on rational quotients. More precisely, let \(q : X\dashrightarrow Q\) be the rational quotient of a compact Kähler manifold \(X\), and let \(L\) be a pseudo-effective line bundle on \(X\) admitting an injection \(L\to(\Omega_X^1)^{\otimes m},~m \geq 1\). We prove that \[ \nu(L,X)\leq\nu(K_Q,Q). \] To the best of our knowledge, both the descent theorem and this inequality are new even in the projective setting.

Since Ilmanen's pioneering work [J. Differential Geom. 38, 417-461, (1993)] it has been a major open problem to understand the sharp-interface limit of systems of coupled Allen-Cahn equations. We prove - for the first time - a global, unconditional convergence result for such a coupled system and show that the limit is a multiphase mean curvature flow in the sense of Brakke.

We develop a unified approach to constructing the moving frames of the evolute, involute and parallel of a Legendre curve via rotational transformations (rotation, boost and null rotation). The definitions of the evolute, involute and parallel of a Legendre curve in hyperbolic and de Sitter 2-spaces are systematically presented. Furthermore, the connections and dualities among these curves are discussed.

We investigate the form of the Saito metric on quotients of the reflection representation of an extended affine Weyl group, restricted to an arbitrary stratum of its discriminant. We compute the Saito determinant for all strata and Dynkin types and show that it is divisible by a product of q-analogues of restricted roots for the associated hyperplane arrangement. Our results simultaneously generalise the factorisation in the Weyl denominator formula to higher codimension strata, and provide q-analogue refinements of results of Antoniou-Feigin-Strachan for Coxeter groups.

It is well known that the Einstein equations are tensor equations for a Lorentzian metric. Hence, choosing a suitable gauge condition is crucial for solving them. As a powerful gauge condition, maximal foliations have played a pivotal role in two groundbreaking works in general relativity: the global stability of Minkowski spacetime and the bounded L^2 curvature conjecture. Nevertheless, if the initial hypersurface fails to be maximal, the usual elliptic approach for constructing maximal foliations encounters fundamental difficulties when solving the Einstein equations. The purpose of the paper is to provide a construction of maximal foliations around any initial asymptotically flat Cauchy surface satisfying vacuum Einstein constraint equations, under a suitable smallness condition on the mean curvature.

We study bubbling for sequences of solutions of the Hamiltonian-perturbed Fueter equation. By Walpuski's compactness theorem, a bounded-energy sequence converges weakly to a Fueter map, with energy loss recorded by a defect measure supported on a codimension-two rectifiable set. Assuming that the limiting map has no non-removable singularities, the bubbling locus contains a nontrivial Lipschitz arc and the bubbles attach to the limiting map along this arc, we show that the limiting Hamiltonian lies in an exceptional subset of infinite codimension. The proof combines quantitative control of rational sweepout loci in hyperkähler manifolds with a transversality argument. We expect both the Lipschitz-arc and attachment assumptions to be automatic. This strongly suggests that, in the absence of non-removable singularities, bubbling of Hamiltonian-perturbed Fueter maps is highly nongeneric.

Let $M$ be an $m (\ge2)$-dimensional closed orientable submanifold in a Riemannian manifold of sectional curvature bounded above by $\delta$. We obtain the extrinsic upper bounds of the harmonic mean of the first $m$ nonzero eigenvalues of the Laplacian on $M$, which extend the previous related results for the first nonzero eigenvalues.

Let $(M,g)$ be a smooth, closed, connected $d$-dimensional Riemannian manifold, and let $X_1,\ldots,X_n$ be i.i.d.\ with common law $f\,d\mathrm{vol}_g$, where $f\in C^4(M)$ is strictly positive. We derive a uniform intrinsic second-order expansion for symmetric three-vertex edge-indicator statistics supported on connected configurations, including the induced-path and triangle kernels. The second-order term separates density variation, normal-coordinate Jacobians, and the curvature-induced motion of the internal-chord boundary. Within this three-vertex connected symmetric class, a universal boundary–moment identity reduces the kernel-dependent contribution to a common intrinsic functional involving $\int_M f\|\grad f\|_g^2\,d\mathrm{vol}_g$ and $\int_M f^3\operatorname{Scal}_g\,d\mathrm{vol}_g$. For the normalized path–triangle contrast, the Euclidean leading term cancels. We construct a consistent estimator of this intrinsic functional and, in a denser bandwidth regime, we prove an exact-expectation-centered root-\(n\) central limit theorem via the first Hoeffding projection. For uniform sampling on a closed surface, the estimator consistently recovers the Euler characteristic. We also study the threshold radius at which the maximum degree of a binomial random geometric graph first reaches two. Using the active-triple intensity expansion and a dependency-graph Poisson approximation, we obtain the order-$n^{-3/d}$ correction to the log-survival law for $d>6$.

We establish the existence and uniqueness modulo time-independent diffeomorphisms of the positively curved ancient Ricci flow $(M^2, \partial M^2, g(t))$ on a two-dimensional surface with boundary, assuming uniformly bounded diameter and constant positive boundary geodesic curvature. In particular, this ancient Ricci flow is rotationally symmetric, its backward limit is the flat disk, and its forward limit is a half-spherical singularity. To our knowledge, this result is the first instance of a classification result for ancient Ricci flows with boundary.

We construct and prove the transversal invariance of two families of pinching cones for the Hamilton ODE $\frac{\mathrm{d}}{\mathrm{d}t}R=Q(R)$ in dimensions $n=7,8$, thereby extending the pinching estimate established by Brendle for $n\geq 12$ and by Chen for $9\leq n\leq 11$. In dimension $n=8$, two steps in Chen's construction genuinely fail. The first, an estimate for the second cone family, is repaired by retaining a coupling discarded in the dimension-general argument. The second is the gluing of the two cone families, for which Chen's estimate fails at $n=8$ by a definite margin. We replace it with a new argument that reduces the gluing to a single linear inequality on the cone of weakly PIC curvature operators, and prove this inequality by an explicit finite combination of frame inequalities. In dimension $n=7$, the two separate cone-invariance results are proved, but the gluing step is not established here. Accordingly, the resulting full pinching estimate and classification statements are restricted to dimension $n=8$. The $n=8$ pinching estimate, together with dimension-free arguments of Brendle, yields the topological classification in dimension eight of compact manifolds with positive isotropic curvature that contain no nontrivial incompressible $(n-1)$-dimensional space forms, extending a theorem of Brendle from $n\geq 12$. Together with curvature-improvement and classification results of Cho–Li and Brendle–Naff, it also yields the classification of noncompact $\kappa$-noncollapsed ancient solutions to the Ricci flow with uniformly PIC, extending a theorem of Cho–Li from $n=4$ or $n\geq 12$.

Let $S\subset\mathbb R^3$ be a properly embedded mean-convex planar surface with finitely many ends. Designate one end as asymptotically flat, assume that $H_S$ is integrable there, and denote its extrinsic mass by $m_+(S)$. Let $A_S$ be the infimum of the areas of compact surfaces separating the distinguished end from all the others. We prove \[ m_+(S)\geq\sqrt{\frac{A_S}{\pi}}. \] No outermost free-boundary minimal surface is assumed, and no asymptotic or integrability condition is imposed on the other ends. If $A_S=0$, equality holds precisely for the Euclidean half-space. The complete catenoid realizes equality with $A_S>0$. Conversely, if equality holds with $A_S>0$, then $A_S$ is attained by a flat free-boundary disk that is outermost toward the distinguished end, and the corresponding component of $S$ is a half-catenoid.

Let $M$ be a connected geodesically complete Riemannian manifold without boundary, write $\mu$ for Riemannian volume, and set $V(o,r)=\mu(B(o,r))$ for geodesic balls centered at $o$. For $0<\alpha<2$, let $X^{(\alpha)}$ be the process obtained by subordinating Brownian motion with an independent $\alpha/2$-stable subordinator; its $L^2$-generator is $-(-\Delta)^{\alpha/2}$. We prove that \[ \int^\infty\frac{dt}{V(o,t^{1/\alpha})}=\infty \] implies that \(X^{(\alpha)}\) is recurrent. The proof uses a spectral trace estimate and radial cutoffs on $M\times(0,\infty)$, where the auxiliary measure is $y^{1-\alpha}\,d\mu\,dy$. It proves the sufficient implication in Grigor'yan's Problem~26.