Papers

In this article, we investigate spectral properties of compact $V$-static manifolds, namely, compact manifolds with boundary whose metrics are critical points of the volume functional under a scalar curvature constraint. We derive sharp estimates for the first Steklov eigenvalue and the entire fourth-order Steklov spectrum. We further obtain a Lichnerowicz-type lower bound for the first eigenvalue of the drifted Laplacian naturally associated with the $V$-static potential. In the corresponding equality cases, we obtain rigidity results characterizing the Euclidean ball and the hemisphere.

We show that a compact mean curvature flow whose singularities have multiplicity-one cylindrical tangent flows admits a smooth Morse resolution. The resolution agrees with the spacetime track outside any prescribed neighborhood of its singular set and all its critical points have the expected index. No nondegeneracy or isolatedness assumption is required. In the mean-convex case the result follows by smoothing the global arrival function.

We introduce a codimension one construction of coarse Ricci curvature in Lorentzian geometry. Using the $1$-Lorentz-Wasserstein distance $\ell_1$, we compare probability measures supported on small spacelike hypersurfaces through nearby events and recover, in a precise asymptotic regime, the ambient Ricci curvature in future-directed unit timelike directions, with a universal dimensional prefactor reflecting this codimension one construction. The construction echoes the Raychaudhuri equation, which relates the evolution of spatial volume expansion along timelike geodesics to Ricci curvature. Our quantitative estimates rely on transport maps with controlled displacement, constructed through a Moser-type flow. At leading order, the slice construction is insensitive to smooth weights. By smearing the spacelike slices into thin timelike tubes and calibrating the temporal and spatial scales, we recover the timelike Bakry-Émery tensor $\mathrm{Ric}+\mathrm{Hess}\,V$ associated with the weighted reference measure $\smash{\mathfrak{m}=\mathrm{e}^{-V}\,\mathrm{vol}_g}$.

As a metric measure space possessing positive Bakry-Émery curvature, a Ricci shrinker with potential function $f$ admits a heat kernel $H_f$ under the weighted volume measure $e^{-f}dv$. In this paper, we study $H_f$ systematically and establish a series of fundamental estimates. Moreover, we clarify the equivalence between $H_f$ and the spacetime heat kernel $H(x,t;y,s)$ under Ricci flow induced by a Ricci shrinker. Inspired by this relation, we extend corresponding analysis tools of Ricci flows to Ricci shrinker metric measure spaces, such as Nash entropy and reduced distance. As a direct application, we prove that $\left|\nabla R\right|=o(f^{3/2})$ implies $R=o(f)$.

Let $S$ be a Kato surface and $D$ its maximal reduced divisor of rational curves. On $S\setminus D$ we construct Hermitian metrics which are flat along the leaves of the canonical foliation and study their evolution under the Chern-Ricci flow. For Enoki surfaces, we construct an immortal normalised solution which, on every compact sublevel of the natural exhaustion, converges in the Gromov-Hausdorff sense to the elliptic base endowed with an explicit flat metric. For Kato surfaces of intermediate type, the affine Green model and its finite-index extension give, under the assumption $0<\mu<2$, an explicit normalised solution which, on every compact Green block, collapses in the Gromov-Hausdorff sense to a circle. In the Enoki case the limiting area is $2\pi b_2(S)$; in the intermediate case the length of the limiting circle is determined by the Green and leafwise monodromies.

We study Hamiltonian stationary Lagrangian surfaces in complex space forms with nowhere-zero mean curvature vector $H$. We prove that if the Gaussian curvature is constant and $\grad|H|$ has positive constant length, then both the surface and the ambient space are flat. Moreover, the immersion is locally congruent to the known surface obtained as the product of a Cornu spiral and its reflection.

We prove that a complete noncompact Kähler surface with nonnegative Ricci and nonnegative quadratic orthogonal bisectional curvature is contractible, and hence homeomorphic to $\mathbb{R}^4$, if it is simply connected at infinity. Under positive bisectional curvature, this removes the contractibility assumption from the conditional uniformization theorem of Datar–Pingali–Seshadri: strong Steinness and simple connectivity at infinity suffice to identify the surface biholomorphically with $\mathbb{C}^2$. We also derive bounded-gradient strictly plurisubharmonic exhaustions and uniform holomorphic kernel estimates for complete $U(n)$-invariant Kähler metrics on $\mathbb{C}^n$ with nonnegative bisectional curvature.

The fact that every smooth knot type admits a Legendrian representative is a classical result in contact topology. However, the analogous surjectivity question was open at the parametric level. In this work we address the $n>1$ case. We prove that for every $n\geq 3$, every knot type $\mathcal K$, every Legendrian representative $\mathcal L$ and every formal Legendrian representative $\mathcal{FL}$, the associated group homomorphisms $\pi_n(\mathcal{L})\to\pi_n(\mathcal{K})$ and $\pi_n(\mathcal{FL})\to\pi_n(\mathcal{K})$ are never surjective. We then show that surjectivity at the $\pi_2$-level depends on the knot type. This work thus proves the presence of rigidity for parametric families at every higher homotopy level beyond $\pi_1$.

We introduce and study branched real projective structures on compact surfaces. Our main result shows that every representation of the fundamental group of a compact surface into $PSL(3,\mathbb R)$ arises as the holonomy representation of a branched real projective structure, regardless of whether the surface is orientable. We also consider the realisation problem with prescribed branching data, relating the branching degree to the second Stiefel–Whitney class of the representation. The results obtained in this direction suggest several interesting avenues for future research.

We investigate the existence conditions for coupled extremal Kähler metrics on minimal ruled surfaces over a genus 2 Riemann surface. Using the Calabi ansatz to reduce the coupled extremal equations to ordinary differential equations, we prove that a pair of coupled extremal metrics exists for normalized Kähler classes $\Omega_a$ and $\Omega_b$ if and only if their parameters $(a, b)$ belong to an explicitly defined open region $\mathcal{S}_{\mathrm{ext}}$ in the positive real quadrant. Furthermore, we show that this existence region inherently contains the diagonal segment corresponding to classical extremal Kähler metrics.

We establish the eventual monotonicity of the Willmore deficit \( \int_{\Sigma}H^2\,d\mu-16\pi \) along the mean curvature flow of smooth closed embedded strictly convex surfaces in \( \mathbb R^3 \), thereby complementing Huisken's exponential convergence with an eventual monotonicity property. As a consequence, the Hawking mass is eventually nondecreasing along the contracting mean curvature flow, in contrast with the usual inclusion monotonicity expected of the Hawking mass. We also discuss totally umbilical solutions in three-dimensional space forms, where the monotonicity directions of the Willmore energy and the Hawking mass depend on the sign of the ambient sectional curvature.

Let $X$ be a genus $g\geq 0$ compact Riemann surface that admits an antiholomorphic involution $\iota$ with non empty fixed-point set, and let $D = \{p_1,\dots,p_k\}\subset \text{Fix}(\iota)$. At each puncture, we prescribe a Scherk map associated to a given real polynomial quadratic differential with even degree and negative leading coefficient. Here a Scherk map is the harmonic diffeomorphism from $\mathbb{C}$ to the interior of an ideal polygon in $\mathbb{H}^2$, whose Hopf differential is the given polynomial. We construct a harmonic map \[ h:X\backslash D \to \mathbb{H}^2 \] whose asymptotic behavior at each puncture matches the prescribed Scherk map. In particular, the image of $h$ tends to an ideal polygon near each end. The resulting $h$ covers harmonic maps obtained by taking the $\mathbb{H}^2$ factors of horizontal catenoids in $\mathbb{H}^2 \times \mathbb{R}$.

Sprays of scalar curvature constitute an important class of sprays, and such a class includes all two-dimensional sprays. In this paper, we consider the global metrizability of certain sprays on a manifold under some curvature and topological conditions. We prove that, for a regular spray G of scalar curvature with almost everywhere nonzero Ricci curvature on a manifold $M$ of dimension $n\ge 3$, G is globally metrizable if and only if it is locally metrizable, provided that $M$ has trivial first de Rham cohomology group. Further, we characterize the global metizability of a class of two-dimensional singular Berwald sprays on a manifold with trivial first de Rham cohomology group. Meanwhile, on a cylinder $S^1\times {\bf R}$ (with non-trivial first de Rham cohomology group), we construct a family of singular Berwald sprays which are locally metrizable but not globally metrizable. Finally, we construct some examples of two-dimensional sprays on cylinders or spheres with special properties.

We prove the uniqueness conjecture on the twisted Kähler-Einstein currents in big cohomology classes, by solving a problem of Bo Berndtsson posed in his work on the uniqueness theorem of Käher-Einstein metrics.

We study when the logarithm $\psi$ of a positive finite exponential sum on $\mathbb{R}^d$ satisfies $\det\nabla^2\psi=C\exp(\langle b,\theta\rangle-\lambda\psi)$. This is the Kähler–Einstein equation for metrics induced by exponential maps into projective space and, for natural exponential families, the condition that the Jeffreys prior be a Diaconis–Ylvisaker conjugate prior. First, we classify the bivariate Laurent polynomials with unimodular support satisfying the generalized Einstein condition of Di Scala and Sombra: up to units and monomial changes of coordinates, they are powers of an affine trinomial or products of powers of two independent binomials. Second, we show in every dimension that a smooth compact toric manifold with a projectively induced Kähler–Einstein metric is a product of projective spaces with matched multiples of the Fubini–Study metrics, immersed by a complete Veronese–Segre system up to automorphisms. This proves the compact toric case of the homogeneity conjecture for such metrics and the fixed-point germ and univalent forms of a conjecture of Manno and Salis. Third, without lattice or rationality assumptions, the finite-support exponential families satisfying the equation are, up to affine changes of statistic, exactly the products of multinomial families with a common ratio of categories to trials; this settles the finite-support case of a question of Casalis.

A $\lambda$-translating soliton is a hypersurface in $\mathbb{R}^{n+1}$ satisfying $H=\langle \mathbf{T},\nu\rangle+\lambda$; equivalently, it has constant weighted mean curvature with respect to the log-linear density $e^{\langle T,X\rangle}$, and is an eternal solution of the mean curvature flow with a constant forcing term. In this paper, we first prove that every complete properly immersed $\lambda$-translating soliton with $\lambda>0$ and $\inf_{\Sigma}H>\lambda$ has at least exponential volume growth, in contrast with the linear growth of ordinary translating solitons. We then prove sharp non-existence results for graphic $\lambda$-translating solitons ($\lambda\geqslant 0$) with bounded gradient, and two rigidity theorems.

We study a quadrature-surface free boundary problem driven by the Riemannian $p$-Laplacian on a smooth compact finite-dimensional Riemannian manifold. We formulate the problem as a shape optimization problem and develop an intrinsic admissible class based on a uniform Riemannian $RC$-$GNP$ condition~\cite{DS3}. We establish compactness of admissible domains and stability of the associated Dirichlet problems under strong $RC$-$GNP$ convergence. In particular, the states converge strongly in $W^{1,p}$ and the associated $p$-torsional energy is continuous. Under additional boundary regularity, we derive the first variation of the shape functional and obtain the free-boundary Euler–Lagrange condition \[ |\nabla_g u_\Omega|_g^p=\sigma H_{\partial\Omega}+k^p. \] We also formulate the contact optimality inequality, prove the Riemannian comparison of second fundamental forms at tangential contact, and obtain a sufficient condition on a reference domain which rules out contact and yields a solution of the free-boundary problem. Explicit radial examples on geodesic balls of the round sphere are included.

In this paper, we study de Rham realizations for finite orientifold quotients. On the orientation preserving inertia of an almost complex global quotient we construct an anti-linear involution of the Chen-Hu algebra whose fixed subalgebra is a real form of the Chen-Ruan algebra. For genuine orientation reversing fixed sectors we prove that no coordinate invariant single sector fractional age exists. When a Calabi-Yau volume line is preserved, the relative phases define a canonical class $[\mu_{\mathbb R}]\in H^2(\widehat G;\mathbb Z_\epsilon)$. We identify this class with the Bockstein obstruction to a coherent real phase lift and prove that its vanishing is equivalent, after rephasing the volume form, to a phase trivial orientifold action. We then formulate a general Euler-Gysin realization theorem using relative Fredholm bundles and determinant-line sewing. For phase trivial Calabi-Yau surfaces we verify the admissibility conditions directly and obtain a canonically defined associative graded cohomology algebra containing the Chen-Hu even sector and the odd fixed surfaces. For an Eisenstein abelian surface with an order six dihedral action we compute the centralizer invariant algebra explicitly; it is a $16$ dimensional Frobenius algebra whose odd sector multiplication is noncommutative in the shifted grading. We also explain the relation with earlier unoriented orbifold cohomology constructions.

Let $\Omega$ be the curvilinear square bounded by the four circular arcs $|\zeta\mp1|=\sqrt{2}$, $|\zeta\mp i|=\sqrt{2}$, and let $W=\sqrt{1+14\zeta^{4}+\zeta^{8}}$. Put \[ \theta(\zeta)=\arcsin\frac{2(1+i)\zeta} {\sqrt{1+4i\zeta^{2}-\zeta^{4}}}, \] \[ f=\tfrac14\bigl(-iF[\theta,\tfrac14]+F[\theta,\tfrac34]\bigr),\quad \quad g=\tfrac14\bigl(\ \ iF[\theta,\tfrac14]+F[\theta,\tfrac34]\bigr), \] \[ \mathfrak h=(2-\sqrt3)\, F\bigl[\arcsin\bigl(i(2+\sqrt3)\zeta^{2}\bigr),(2-\sqrt3)^{4}\bigr]. \] These are, verbatim, the three functions of our companion paper $\mathrm{[D]}$ on the diamond surface $\mathrm{D}$: the two papers start from the same point, and differ only in which real parts are taken. With $X=(\kappa\,\operatorname{Re}f,\ \kappa\,\operatorname{Re}g,\ \tfrac12+\kappa\,\operatorname{Im}\mathfrak{h})$ and $\kappa=3/(2K[1/9])$, the map $X$ parametrizes a fundamental patch of Schwarz's primitive surface $\mathrm{P}$, the conjugate of $\mathrm{D}$. Part I builds the algebraic apparatus $\ldots$ the normalizing constant is identified exactly: $\varpi:=\kappa^{-1}=K[-3]=\tfrac12 K[3/4]=\tfrac23 K[1/9]$. Part II carries out the elimination and completes the proof of \[ (\star)\qquad \operatorname{sn}\bigl(\varpi(x{+}y),-3\bigr)\,\operatorname{sn}\bigl(\varpi(x{-}y),-3\bigr) =\tfrac13\operatorname{sn}\bigl(3\varpi(z-\tfrac12),\tfrac19\bigr). \] Part III shows that $(\star)$ is, after all, additively separable $\ldots$ for an overview of this paper and $\mathrm{[D]}$, please see arXiv:2609.14206.

We study umbilic slopes and the global classification of cubic Weingarten surfaces. For a smooth surface in Euclidean three-space, a nonconstant principal curvature germ with a unique secant tangent has slope zero, minus one, infinity, an odd integer at least three, or its reciprocal. At a nonconstant umbilic germ, a smooth regular relation between mean and Gaussian curvature supplies such a tangent after a continuous labeling of the principal curvatures; the ordered curvature image has at most two limiting directions, both in the same set. We also classify compact immersed surfaces without boundary of class $C^4$ satisfying $\kappa_2=c\kappa_1^3$ in some order at every point, for a fixed real $c$: each connected component is embedded as an ellipsoid of revolution. The arguments share a classification of homogeneous Hessian identities at an umbilic. Scalar unique continuation treats infinite-order contact in the smooth slope problem, whereas an exact fourth-order growth estimate and angular energy estimates yield the cubic classification.

We study the radical and screen distributions of a lightlike hypersurface of an indefinite statistical manifold with respect to the induced connections arising from the ambient dual affine connections. Necessary and sufficient conditions are obtained for each distribution to be parallel with respect to both induced connections. These conditions are expressed through the screen shape operators, the local screen fundamental forms, the mean induced connection, and the tangential difference tensor. When both distributions are parallel, we characterize the screen projection and obtain a block decomposition of the tangential difference tensor, whose mixed radical-screen part vanishes. We also obtain equivalent conditions for the induced connections to be metric connections. In the metric case, the induced screen connections coincide with the Levi–Civita connection on each integral manifold of the screen distribution, and the curvature tensors of the three induced connections agree on screen directions. We further determine the action of these curvature tensors on the radical direction. An explicit example shows that both distributions may be parallel although neither induced connection is a metric connection and their curvature tensors need not vanish.

We find Sasaki-Einstein metrics on rational homology $(4n-1)$-spheres for $n>1$ built from cyclic polynomials of index 1 cutting out rational varieties. The Einstein metrics found here are inequivalent to the ones found by Boyer and Galicki in arXiv:math/0311355. Our findings are consequence of an improvement, for hypersurfaces defined by cycle polynomials, on the estimate given by Johnson and Kollár to determine Kähler-Einstein orbifold metrics. We also construct weighted hypersurfaces that contain the rational varieties described above as codimension two subvarieties and, due to the refined estimate for cyclic polynomials, we find conditions on the weights and degrees of these hypersurfaces so their corresponding smooth links admit Sasaki-Einstein metrics. Finally we study the effect of the Berglund-Hübsch transpose rule on the topology and on the existence of Sasaki-Einstein metrics on the links studied and generalize all the results given in arXiv:2311.15998 for rational homology 7-spheres to rational homology $(4n-1)$-spheres, that is, we show invariance of these two features under the transpose rule.

We prove sharp upper and lower bounds for the Morse index and nullity of linearly full branched minimal 2-spheres in the round 2n-sphere. In particular, we show that the Morse index of a linearly full minimal 2-sphere in the round 2n-sphere is at least n(n-1)(2n+1), with equality if and only if the 2-sphere is twistor-equivalent to the Boruvka sphere. Our techniques also apply more generally to give bounds for the Morse index and nullity for totally isotropic surfaces in the 2n-sphere.

The completeness properties of spaces of immersed curves endowed with Sobolev-type reparametrization-invariant Riemannian metrics have been studied extensively in recent years. Whereas constant-coefficient metrics of large enough order on the space of closed curves are known to be complete, this is not the case for open curves. In this work, we characterize the metric completion of the space of real-valued immersed open curves with respect to these metrics. We show that the set of additional limit points in the completion is homeomorphic to $H^n$-diffeomorphisms of a closed interval. This result answers a question of the structure of the completion, posed in (Bauer et al., Ann. Sc. Norm. Super. Pisa Cl. Sci. (2023)), in the negative, and suggests a more refined conjecture on the structure in the case of multi-dimensional ambient space.

We present a framework for the modeling and analysis of geometric hybrid dynamical systems as hybrid inclusions on $C^1$-manifolds. Using tools from nonsmooth and set-valued analysis on manifolds, we derive coordinate-independent sufficient conditions for the existence of nontrivial solutions to this type of systems. We present geometric notions of uniform stability and attractivity of compact sets, and establish their equivalence to metric-based stability notions when the manifold is endowed with a Riemannian structure. We also introduce nonsmooth Lyapunov functions and a hybrid Lyapunov theorem providing sufficient conditions for uniform global asymptotic stability of compact sets. Finally, by characterizing $\omega$-limit sets of precompact solutions, we derive a hybrid invariance principle for geometric hybrid dynamical systems. The results are demonstrated through several running examples.

We prove that every smooth properly embedded free-boundary minimal annulus in the unit ball is congruent to the critical catenoid. We also classify positive solutions of $(\Delta_{\mathbb{S}^2}+2)u=0$ on smooth spherical annuli with $u=0$ and $|\nabla u|=1$ on the boundary; their one-homogeneous extensions are the axially symmetric Alt-Caffarelli cones. Lastly, we also treat the case of free-boundary minimal annuli in spherical caps.

For a real analytic map germ $G:(\mathbb{R}^m,0)\to(\mathbb{R}^p,0)$, we study Milnor fibrations obtained by replacing the squared Euclidean distance with an analytic control function $\rho$ defining the origin. We formulate the corresponding Milnor set and condition (b), derive criteria for tube and sphere fibrations, and examine their dependence on $\rho$. The family $G_\rho=(xz,\,yz\rho)$ provides explicit changes of regularity under elliptic controls. In particular, the germ $G(x,y,z)=(xz,\,yz(10x^2+y^2+3z^2))$ admits neither the induced tube nor sphere fibration for the Euclidean control, while both fibrations exist for the adapted function $\rho=10x^2+y^2+3z^2$. Thus the control function can be essential to the local fibration structure.

We prove a purely quasi-local positivity theorem for the Wang–Yau mass for a class of initial data whose Jang deformation, after a boundary-preserving conformal reduction to zero scalar curvature, lies in a sufficiently small transverse–traceless (TT) perturbative neighborhood of a strictly convex Euclidean fill-in.We explicitly construct a nontrivial class of physical initial data whose admissible Jang reductions realize this TT-generated sector. The argument reduces the Wang–Yau energy to the Brown–York mass of the resulting scalar-flat compact metric, together with nonnegative bulk terms determined by the Jang deformation, and establishes strict positivity by computing the second variation of the Brown–York functional at the Euclidean metric in transverse–traceless directions. The proof is entirely confined to the compact fill-in and uses neither an asymptotically flat extension nor the positive mass theorem. This gives a partial answer to a question of R. Schoen concerning a genuinely quasi-local proof of positivity for quasi-local mass.

We consider an application of the Atiyah-Bott fixed point theorem in quaternionic geometry. We review the relation of the general Salamon complex of a quaternionic manifold and the Dolbeault complex with coefficients in a certain holomorphic vector bundle of its twistor space. Furthermore, we investigate which maps are suitable for the application of the fixed point theorem. As a result, we obtain a fixed point formula for both complexes and apply it to Wolf spaces.

We develop an algebraic mechanism based on Laplacian algebras for constructing minimal hypercones and determining their variational properties. Generator counts and degrees organize the constructions; homogeneous multipliers depending on the generators yield subcalibration fields, stability bounds and quantitative area-minimizing inequalities. Two-generator isoparametric algebras give polynomial subcalibrations for their minimizing hypercones. In ambient dimension at least three, strict gradient-square closure forces nonlinear odd degree to be cubic. We then study algebras of degrees $(2,2,g)$ and quadratic compositions, placing OT–FKM splitting quartics and Riedler's pullbacks in a common quadratic-composition framework. It yields two further infinite quartic families with a sharp minimizing threshold, as well as harmonic sextics from complex Gram coordinates. For an irreducible harmonic minimal cubic on its essential space, we prove a sharp bound of five generators for its smallest Laplacian algebra. We determine the minimizing range of Clifford cubics and construct irreducible nonradial harmonic examples in ambient dimensions $8,14,26$; the last is strictly area minimizing. Finally, spectral families whose smallest algebras have arbitrarily many generators provide examples in every degree at least two: real parts of octonionic products, complex Pfaffians and complex determinants. Explicit potentials establish strict area minimality and stability bounds, including across nonisolated singularities. The algebraic identities and certificates are supported by reproducible exact symbolic computations.