Papers

We show that the Wasserstein space of the circle is not flat, in contrast with the flatness of the Wasserstein spaces of the real line and of an interval. Its sectional curvature is governed by a symplectic form on the Fourier coefficients, so that a plane is flat exactly when it is isotropic for that form, and on the plane spanned by the $n$-th Fourier modes the curvature is bounded above by $3n^2$, with equality exactly at the uniform measure. It is therefore unbounded. We compute it inside the formal Riemannian structure of Otto and confirm it directly from the definition of $W_2$, which also shows that no neighbourhood of the uniform measure in $P_2(S^1)$ is CAT(0). The result corrects the widely stated expectation that a Wasserstein space is flat whenever its base manifold is one-dimensional. The computation is carried out in the global frame of the tangent bundle of $P^\infty(M)$ induced by the eigenfunctions of the Laplacian of $M$, whose existence was observed by Lott. We prove that it is a Schauder basis of every tangent space and write Otto's metric and its Levi-Civita connection in it. On the circle this gives the metric and connection coefficients of $P^\infty(S^1)$ in closed form, as finite combinations of the Fourier coefficients of the density. A final section puts the frame to work on a computation from dynamics, reading the derivative of the push-forward action of an expanding circle map at its invariant measure as a weighted shift on the Fourier frequencies.

We establish a Pogorelov type interior $C^2$ estimate for the real twisted Monge-Ampere equation without assuming uniform ellipticity. This equation is fully nonlinear and elliptic, but is neither concave nor a function of the eigenvalues of the Hessian, so the standard techniques for interior second-derivative estimates do not directly apply. Building on the partial Legendre transform formulation of Streets and Warren, we derive a differential identity for the associated positive-definite matrix $W(D^2u)$ and use some linear algebra to control the third-order terms arising in the maximum principle argument. As an application, we prove a Liouville theorem for entire split convex solutions with quadratic growth, extending the rigidity theorem of Streets and Warren by replacing their uniform ellipticity hypothesis with suitable quadratic growth assumptions.

Smyth (1984) proved that every tetrahedron $T\subset\mathbb{R}^3$ contains three embedded minimal disks meeting $\partial T$ orthogonally, obtained as conjugate surfaces of Plateau solutions in a skew quadrilateral $Q$ whose edges are $c_i\nu_i$, where $\varphi_1,\varphi_2,\varphi_3,\varphi_4$ are faces of $T$, $c_i=\operatorname{Area}(\varphi_i)$ and $\nu_i$ are outward normal vectors. For the $\textit{regular}$ tetrahedron we give the Weierstrass data explicitly. The Gauss map is a Schwarz triangle function for a circular-arc triangle with angles $(\pi/4,\ \pi/2,\ \pi\beta/2)$, where $\beta=\arccos(-1/3)/\pi$; the hypergeometric parameters are linear in $\beta$ and there is $\textit{no accessory parameter and no free parameter}$. We record a closed form for $\kappa$ in $\Gamma$-values, a Wronskian closed form for $g'$, the exact total curvature $2\pi(2\beta-1)$, the symmetric-square structure of the data, and a rigidity statement: since $\beta/2$ is irrational, the projective monodromy is dense and the data $\textit{cannot}$ be algebraic. Numerically the representation extends to 28 digits – $\operatorname{Area}/\ell^{2}=0.2172341554075574483760009126\ldots$ – confirming a 2013 Surface Evolver computation of Ken Brakke.

We study the marked analytic moduli space of polystable degree-zero $\mathrm{SL}_r(\C)$-Higgs bundles over Teichmüller space. The connected components of the orbit-type loci defined by the full determinant-one stabilizers form a complex Whitney stratification. The projection to Teichmüller space is a holomorphic submersion on every stratum. Each stratum carries a canonical relative holomorphic symplectic form. These forms determine a vertical holomorphic Poisson bracket on the reduced structure sheaf, whose symplectic leaves are the connected orbit-type strata in the fibers. On each fiber, the Hitchin–Kobayashi correspondence is biholomorphic on every stratum and identifies its hyperkähler structure. The harmonic representatives and vertical hyperkähler tensors vary smoothly along the strata of the total space.

We establish numerical criteria for polynomial positivity of real $(1,1)$-classes on compact Kähler manifolds for strictly right-Noetherian polynomials. As applications, we obtain existence and uniqueness for the generalized Monge-Ampère equations with a smooth zeroth coefficient, the complex Hessian and Hessian quotient equations, and the critical LYZ equation. We also characterize the cone of $k$-positive classes as a connected component of a numerical positivity cone.

We show that compact locally symmetric manifolds $M$ with universal cover the symmetric space $X$ for $SL(n,\mathbb{R})$ form a topological higher $d$-expander family for $d\leq n/8$. We prove the same statement for $SL(n,\mathbb{R})$ replaced by a split simple non-compact real Lie group $G$ and for $d$ linear in the rank of $G$. We accomplish this by showing that minimal submanifolds of low codimension in such $M$ must have volume comparable to the volume of $M$. Our proof is based on a new monotonicity formula for minimal submanifolds of $X$, together with bounds on the decay of matrix coefficients for unitary representations of higher rank Lie groups. We also give the first locally symmetric example of power-law systolic freedom. This paper partially supersedes \cite{fl24}.

We compute the Morse index of several complete nonorientable minimal surfaces immersed in Euclidean three-space. In particular, we prove that the Meeks minimal Möbius band has Morse index two, which is the least possible Morse index for such nonorientable minimal surfaces, by previous works of Ros and Chodosh-Maximo. This is the first known minimal surface with index two. We show that the López minimal Klein bottle has Morse index three. As a consequence of this computation, we extend results in the literature and observe that any complete minimal surface immersed in Euclidean three-space with total curvature $8 \pi$ has Morse index three. We also compute the Morse index of the Oliveira family of minimal Möbius bands in terms of their total curvature and show that every non-negative integer is the Morse index of a complete minimal surface immersed in Euclidean three-space. We show that five is a sharp lower bound for the Morse index of any minimal oriented double cover in Euclidean three-space, and equality is attained by the double cover of the Meeks minimal Möbius band. Finally, we prove results towards the classification of complete minimal surfaces with Morse index two immersed in Euclidean three-space, obtaining topological and geometric restrictions on such surfaces.

For any pseudo-Euclidean space $V \simeq \mathbb{R}^{p, q}, p \geq 3$ and module $W$ over the even Clifford algebra $\text{Cl}^{0}(V)$, V. Córtes constructed noncompact homogeneous quaternion pseudo-Kahler space. In particular, such homogeneous space is always Einstein and for $p=3$ it is Riemannian. Based on this approach we construct the series of Riemannian Einstein homogoneous spaces with negative scalar curvature associated with $V\simeq \mathbb{R}^{p, q}, p\equiv 3 \pmod{4}$ and an irreducible $\text{Cl}^{0}(V)$-module $W$.

We investigate Lie groups endowed with left-invariant Lorentzian metrics of nonzero constant sectional curvature. We first revisit Heintze's theory and obtain a characterization of the Lie algebras of Riemannian Lie groups of negative constant curvature. We also extend classical results of Nomizu and Barnet to the pseudo-Riemannian setting by constructing a large family of Lie groups carrying incomplete left-invariant metrics of constant sectional curvature. We then describe Lorentzian Lie algebras of nonzero constant curvature according to the causal nature of their center and derived ideal. This description is complete except when the derived ideal is Lorentzian, for which we obtain a complete classification in dimension four. We show that every left-invariant Lorentzian metric of nonzero constant curvature on \(\mathrm{SL}(2,\mathbb R)\) is bi-invariant and complete. Moreover, a semisimple Lie group admits a complete left-invariant Lorentzian metric of nonzero constant curvature if and only if it is locally isomorphic to \(\mathrm{SL}(2,\mathbb R)\). We also characterize \(\mathrm{SL}(2,\mathbb R)\) through the existence of a noncentral spacelike left-invariant Killing vector field. Finally, we classify Lorentzian Lie algebras of nonzero constant curvature in dimensions at most four.

We study the inversion of the geodesic ray transform from finitely many local averages of its data. Under suitable geometric assumptions, we prove Lipschitz stability on any fixed finite-dimensional reconstruction space when the measurement partition is sufficiently fine. We develop convergent reconstruction algorithms based on Steepest Gradient Descent and Conjugate Gradients and implement them using piecewise constant finite element spaces. Numerical experiments in 2D and 3D illustrate the influence of geometry and measurement discretization on reconstruction quality.

We show that the space of vacuum initial data sets on a closed manifold often has many non-trivial homotopy groups. The starting point is a result of the second named author, which constructs non-trivial elements in the homotopy groups of the space of initial data sets satisfying the strict dominant energy condition, together with a later result in joint work with Bernd Ammann showing that these elements often persist when the strictness assumption is dropped. In this work, we use a parametrized version of the conformal method to show that these elements may also be represented by maps into the space of vacuum initial data sets. This requires two results that may be of independent interest: metrics admitting conformal Killing vectors can be removed from the space of metrics without changing its weak homotopy type, and over the remaining metrics the York decomposition can be carried out in families, the TT-tensors forming a trivial Hilbert bundle. To our knowledge, this is the first result on the global topology of this space.

We construct non-flat non-Hermitian Yang–Mills connections with zero Einstein constant on smoothly trivial rank-two bundles over compact Kähler surfaces, with positive harmonic metrics and stable induced and adjoint holomorphic bundles. Thus vanishing Chern classes do not force flatness even under stability on both sides. The local model comes from a known complex anti-self-dual ansatz; the stable descents yield explicit global moduli phenomena. For a fixed stable bundle, the Kaledin–Verbitsky map contracts a complex line containing flat and non-flat points in the connected component of the Hermitian–Einstein connection. Contraction also occurs in the two-sided stable locus. The pair of holomorphic projections has a fibre containing both flat and non-flat points. An energy identity and a spectral-gap estimate quantify local flatness with the induced holomorphic structure fixed.

We construct smooth, non-locally-conformally-flat metrics on the closed ball for which the boundary Yamabe equation admits $L^\infty$-unbounded sequences of positive solutions. The background metric and the prescribed scalar and boundary mean curvatures are fixed along each sequence. For zero scalar curvature and positive boundary mean curvature, such examples exist with umbilic boundary in every dimension $N\ge22$ and with nonumbilic boundary in every $N\ge15$. For positive scalar curvature and minimal boundary, the corresponding ranges are $N\ge21$ and $N\ge15$. For every $N\ge9$, examples with either umbilic or nonumbilic boundary also exist when the scalar curvature is $N(N-1)$ and the prescribed boundary mean curvature is below a negative threshold depending on $N$. The construction uses polynomial metric perturbations and corrected bubbles on the half-space, followed by conformal compactification to the ball. We evaluate or estimate the correction solving the linearized Neumann or Robin boundary problem and its contribution to the quadratic term of the reduced energy. We prove that this quadratic term has negative, nondegenerate local extrema with respect to tangential translations and scale in every stated dimension.

Let $Z$ be a compact Kähler manifold endowed with a Hamiltonian action of a compact connected Lie group $U$, and let $G\subset U^{\mathbb C}$ be a real compatible subgroup. Using the gradient map $\mu_{\mathfrak p}$ associated with the momentum map, we establish a Calabi-Matsushima type decomposition for the Lie algebra of the stabilizer $G_z$ at a critical point of $f\circ\mu_{\mathfrak p}$, where $f$ is the restriction to $\mathfrak p$ of a suitable $Ad_U$-invariant strictly convex function on $i\mathfrak u$. More precisely, we show that the isotropy algebra decomposes into eigenspaces corresponding to nonnegative eigenvalues of an adjoint endomorphism, with zero eigenspace given by its reductive part. This extends the classical decomposition for the complexified action of $U^{\mathbb C}$ to the real reductive setting. As an application, we prove that the identity component of the compact stabilizer at critical points of $f\circ\mu_{\mathfrak p}$ is a maximal compact subgroup of the identity component of the $G$-stabilizer.

Landmark shape spaces arise from configurations of $n$ distinct landmarks with rigid motions factored out and scale normalized. When equipped with sufficiently regular metrics inherited from right-invariant metrics on diffeomorphism groups, the shape spaces inherit geodesic completeness from geodesics of diffeomorphisms. We now answer the corresponding stochastic question if the Riemannian Brownian motion on landmark shape spaces exists for all time and hence rule out initially distinct landmarks colliding when following a Brownian flow. We show that with general classes of metrics, including degenerate metrics with infinitesimal rigid motions in their null space, landmark shape spaces are stochastically complete, thus making the use of Riemannian Brownian motion for modelling shape stochasticity in applied fields well-founded.

We prove that a compact immersed disk of constant mean curvature in $\mathbb S^3$ or in $\mathbb H^3$ whose boundary is mapped diffeomorphically onto a circle is an embedded totally umbilical disk.

The known proofs of the gradient estimate for complex Hessian equations rely on the second order estimate of Hou-Ma-Wu and on a Liouville theorem of Dinew-Kolodziej. In this paper, we give a direct proof of the gradient estimate for concave fully nonlinear elliptic equations on compact Hermitian manifolds, which uses neither the Hou-Ma-Wu estimate nor the Liouville theorem. Instead, the proof combines a stability estimate with a comparison argument against nearby smooth admissible functions. We also give an independent proof of the complex Hessian estimate on compact Kähler manifolds, based on Dirichlet Green's functions, under additional assumptions on the operator.

We determine the exact number, up to rotations, of nonconstant positive $C^2$ solutions to the planar isotropic $L_p$ dual Minkowski problem for every $(p, q) \in \mathbb{R}^2$. We obtain a new parametrization of the associated period integral, characterize the regions where the period is strictly increasing or strictly decreasing, and prove that in the remaining nonmonotone region it has a unique nondegenerate maximum. As a consequence, we have a complete classification.

We establish the Type I estimate for the scalar curvature of the Kahler-Ricci flow on compact Kahler manifolds, for arbitrary smooth initial metrics, whenever the flow develops finite time singularities. This estimate is a building block for the analytic minimal model program with Ricci flow.

For immersions of a closed $2h$-dimensional manifold $\Sigma$ into $\mathbb R^m$, we consider curvature energies that are invariant under ambient isometries and dilations, with the principal term $\int_{\Sigma} \big|\nabla^{(h-1)} \vec{\mathrm{I\!I}}\big|_g^2\,d\text{vol}_g$ and lower-order submanifold invariants. Following the four-dimensional approach developed in joint work with Bernard, Martino, and Rivière, we prove that every weak critical immersion in the natural Sobolev class $W^{h+1,2}$, whose induced metric and its inverse have $L^\infty$ coefficients, is real-analytic in harmonic coordinates. The proof combines geometric conservation laws, structural identities, and elliptic estimates with critical Sobolev coefficients to obtain Morrey decay and bootstrap to full regularity.

We consider objects in the Fukaya category of a class of K3 surfaces defined by certain Lagrangian spheres. Assuming that homological mirror symmetry for K3 surfaces holds with sufficiently strong (expected) properties, we prove that, in this case, stability with respect to a suitable Bridgeland stability condition implies the existence of an isomorphic special Lagrangian sphere, as predicted by the general Thomas-Yau-Joyce conjectures. In particular this holds unconditionally for suitable quartic surfaces in $\mathbb{P}^3$ or sextics in $\mathbb{P}(3,1,1,1)$ and their mirrors. A variant holds on the Calabi-Yau threefolds obtained by taking the product of our K3 surfaces with an elliptic curve. These seem to be the first results of this type on compact manifolds.

Naff's planarity estimate is a crucial tool for analysing singularities of quadratically pinched mean curvature flows in high codimension, showing that high-curvature regions become asymptotically codimension one. Naff established the estimate throughout the full Andrews–Baker pinching range in dimensions at least $7$, while in dimensions $5$ and $6$ a stronger pinching assumption was required. We close this remaining gap, proving the planarity estimate in dimensions $5$ and $6$ throughout the full Andrews–Baker range. The key new ingredient is a sharp Kato-type inequality relating the twisting of the principal normal to the gradient of the second fundamental form.

An old conjecture in non-Kähler geometry states that if the Chern holomorphic sectional curvature of a compact Hermitian manifold is equal to a constant $c$, then the metric must be Kähler when $c\neq 0$ and be Chern flat when $c=0$. The conjecture is known to be true in dimension two by the work of Balas–Gauduchon and Apostolov–Davidov–Muškarov in the 1980s and 1990s. Recently, Qin and Tian proved the conjecture in complex dimension three when $c\neq 0$, and the $c=0$ case was proved by Chen–Li under the additional assumption that the metric is balanced. In this article we remove this additional hypothesis and complete the confirmation of the conjecture in complex dimension three. The proof relies heavily on the fact that in dimension three the torsion $3$-tensor can be equivalently expressed as a $2$-tensor twisted by the canonical line bundle, plus Bochner type integration formulas. In particular, the method cannot be directly generalized to higher dimensions.

For a map $F:\Omega\subset\mathbb{R}^n\to\mathbb{R}^m$, let $\Theta(F)$ be the $n$-form on $\Omega\times\mathbb{R}^m$ introduced by Tsai and Wang [arXiv:2604.04336]. If $\lambda_1,\ldots,\lambda_{r}$ are the nonzero singular values of $d F$ at any fixed point, we prove that $$ \Theta(F) \text{ has comass one if and only if }~ \mathcal{S}(\lambda):=\sum_{i=1}^{r}\frac{\lambda_i^2}{1+\lambda_i^2}\leq1. $$ Note that if $\text{rank} d F\le1$, $\mathcal{S}$ is always less than $1$. We also prove a dichotomy that if $\mathcal{S}\leq 1$, then either $\mathcal{S} < 1$ or $\mathcal{S}\equiv1$. When $\mathcal{S}<1$, the graph tangent plane is the only calibrated plane, generalizing the corresponding result for hypersurfaces. When $\mathcal{S}\equiv1$, we prove that $F$ is affine or of rank $2$.

In this article, we study the asymptotic expansion of the analytic torsion associated with positive Hermitian line bundles on projective spaces. We show that the expansion contains no terms of the form $k^{-i}\log k$ for $i\geq 1$. The proof combines the arithmetic Riemann-Roch theorem with an explicit computation of the arithmetic degree of spaces of global sections.

Inspired by the Green-function monotonicity formulas of Colding, Minicozzi, and Manea, we establish weighted monotonicity formulas on closed weighted Riemannian manifolds satisfying ${\rm Ric}_f^m \ge (m-1)kg$, where $m>n\ge 3$ and $k>0$. Our results extend Manea's elliptic monotonicity framework to the setting of finite-dimensional Bakry-Émery curvature and provide a positive-curvature counterpart to the weighted formulas of Song-Wei-Wu. Starting from the Green function of $-\Delta_f+m(m-2)k/4$, we derive an exact weighted Bochner identity and prove three monotonicity formulas, together with a one-parameter family for $\beta\ge (m-2)/(m-1)$, under explicit pole-integrability conditions. We also treat the case of negative effective dimension $m<0$, for which the nonnegative square decomposition remains valid, while the change in the behavior near the Green pole leads to reversed area and volume monotonicity on finite sublevel sets. We conclude with comparisons and questions concerning entropy monotonicity formulas and positive-curvature RCD spaces.

Following the recent work of M. Womack \cite{Wo26} we consider the steady Ricci Yang-Mills solitons on 2d surfaces containing orbifold point $\mathbb{R}^2/\mathbb{Z}_p$. We show that there are a family of solitons depending parameter $\lambda$ which approaches to $\mathbb{Z}_p$-quotient of Hamilton cigar soliton as $\lambda \to (-2/p)^+$ and approaches to (after rescaling) $\mathbb{Z}_p$-quotient of round sphere as $\lambda \to \infty$. For any integer $q >p$ there is a $\lambda$ whose corresponding soliton metric is defined on football orbifold $S_{p,q}^2$.

A classical 1977 result by Roter states that a Riemannian manifold of dimension at least four with parallel Weyl curvature must be locally symmetric or conformally flat. We establish Roter's theorem in the symplectic category: namely, a Fedosov manifold with parallel symplectic Weyl curvature must be locally symmetric or Weyl-flat.

The Gaussian curvature of the graphs of cubic polynomials $f(z)=(z-a)(z-b)(z-c)$, with triangle of zeros $\triangle=\triangle(a,b,c)$, is analyzed. We introduce the composed Pythagorean mean $\mathtt{B}_{\triangle}$, a combination of the three classical Pythagorean means, which plays a central role in the description of the critical points of the Gaussian curvature. When $\triangle$ is not equilateral, the curvature has exactly five critical points: a global maximum at the barycenter, two local minima, and two saddle points, forming a rhombus whose vertices lie on the axes of the Siebeck-Marden ellipse of the triangle. We also initiate a comparative study of $\mathtt{B}_{\triangle}$ with the skew of the triangle by describing a family of polynomials whose zeros determine a specific family of isosceles triangles. The cubic case provides a concrete model strongly suggesting that the critical geometry of the Gaussian curvature of the graph can reveal both analytic and algebraic properties of the underlying holomorphic function.

In this note we present a direct relation between the geometrical definition of a quasi-Einstein metric and Horava gravity: the solutions of the large-distance effective action of Horava gravity that are static and orthogonal are quasi-Einstein metrics. The converse is also true if the function over the quasi-Einstein manifold satisfies a given equation.