Papers

We show that any Serrin domain in $\mathbb{R}^2$ with two boundary components is a member of the canonical unduloidal family

We classify all complex generalized spin-c Killing spinors with complex Killing function $i\mu-\alpha$ on complete Riemannian surfaces, presenting an explicit list of all possible cases. We show that, contrary to previous statements in the literature, there exist complex generalized spin-c Killing spinors on Riemannian surfaces for which $\alpha\mu\not\equiv0$. We illustrate the classification by providing several explicit local, compact, and complete examples, including a complete example for which $\alpha\mu>0$ everywhere. We then move to three dimensions, proving that a spin-c Riemannian three-manifold admits a complex generalized spin-c Killing spinor with complex Killing function $i\mu-\alpha$ if and only if it is a $\mu$-exact trans-Sasakian manifold of type $(\alpha,\mu)$. In addition, we construct a geodesically incomplete Riemannian three-manifold equipped with a complex generalized spin-c Killing spinor for which neither $\mu$ nor $\alpha$ vanishes identically, and then prove that this is not possible if the characteristic vector field of the associated trans-Sasakian structure is complete. These results settle a conjecture by Große and Nakad (Results in Mathematics, 2015) stating that on two- and three-dimensional Riemannian manifolds the complex Killing function of any complex generalized spin-c Killing spinor must be either globally real or globally imaginary.

We prove the spacetime Penrose inequality for asymptotically flat initial data sets $(M^n,g,k)$ in all dimensions $n\geq 3$, under the dominant energy condition, for data admitting the 'maximally parameterized form' defined in equation (4.7). In this case, the second fundamental form $k$ is locally parameterized by $n(n+1)/2$ functions, giving the same formal local functional degrees of freedom as an arbitrary symmetric $2$-tensor. The proof is based on a new 1-parameter family of generalized Schoen-Yau identities and their gauge-invariant specialization, inspired by the generalized Jang framework of Bray-Khuri [8, 9]. For maximally parameterized data, these identities produce a Riemannian comparison metric \[ \bar g=g+\beta\otimes\beta \] with nonnegative scalar curvature $R_{\bar{g}} \geq 0$. Under a natural nondegeneracy condition (5.7) at the apparent horizon, the comparison metric admits a smooth minimal completion up to $\partial M^n$. Together with suitable asymptotic control, this reduces the Penrose inequality to its Riemannian counterpart. The all-dimensional conclusion uses the Riemannian Penrose theorem of Bi-Zhu [6], together with a minimizing enclosure argument that allows singular boundaries. More generally, in the spirit of [8, 9], we note that the Penrose conjecture follows whenever a certain system of PDEs (3.18) involving $k$ can be solved, in which case our new identity (3.15) immediately implies $R_{\bar{g}} \geq 0$. A natural problem is thus to regularize our degenerate system to prove a general existence theory, thereby proving the Penrose conjecture. Essentially, we believe this is what the OpenAI paper [27] posted last week on October 6, 2026 claims to have done.

We study disjoint solution pairs $(u,D)$ to the minimal surface equation in $\mathbb{R}^n$, with $u=0$ on $\partial D.$ A conjecture of W. Meeks states that at most two such pairs can exist in $\mathbb{R}^n.$ We confirm the conjecture in two complementary cases. For $n\in\{2,3,4,5,6\},$ we prove it under a superlinearity assumption. In dimension $n=2,$ we establish the result under a suitable sublinearity assumption. We also derive estimates on the asymptotic angle of domains supporting nontrivial solutions for $n\geq 3.$

We prove that every mod $k$ integral cycle $T$ of sufficiently high codimension in an oriented Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by a cycle which is a $\mathbb{Z}_k$-smooth submanifold $\Sigma$ of nearly the same area, up to a singular set of codimension 5. If in addition $T$ belongs to a mod $k$ homology class admitting an embedded $\mathbb{Z}_k$-smooth manifold representative, then $\Sigma$ can be taken fully regular, up to the codimension one stratum of classical singularities. This extends the smooth approximation theory to cycles with coefficients in $\mathbb{Z}/k\mathbb{Z}$ in stable regimes. Moreover, we provide examples showing the impossibility of locally approximating minimizing cycles by smooth or $\mathbb{Z}_k$-smooth submanifolds, even in homology classes with smooth or $\mathbb{Z}_k$-smooth representatives.

We prove a maximal diameter rigidity theorem for closed 3-dimensional Riemannian manifolds under a spectral lower bound for the Ricci curvature. We show that if $0<\gamma<2$, $K>0$ and $\lambda_1(-\gamma\Delta+\operatorname{Ric})\geq 2K$, then the optimal bound $\operatorname{diam}(M,g)\leq\frac{\pi}{\sqrt{K}}$ is attained if and only if (M,g) is isometric to the round 3-sphere of constant sectional curvature $K$.

We prove a new extension principle for the spherically symmetric Einstein–Maxwell–charged scalar field system. Rather than imposing conditions for mass ratio $2m/r$ as in Christodoulou's small-mass-ratio criterion or as in An–Chen–Singh's proof [1] with $2m/r<3/8$, here we work through a proposal by An-Tan [2] with finite Doppler exponent (blueshift). By conducting two successive blow-up analyses, followed by a stationary comparison and a rigidity argument, we establish the uniform boundness of $\alpha'$ with $\alpha'=\partial_v\alpha/\partial_vr$ and $\alpha=\partial_v(r\phi)/\partial_vr$. This provides the desired extension principle in An-Tan [2] and completes the picture of weak cosmic censorship developed therein. As we were about to release our preprint, Cicortas communicated to us that together with his collaborators, they concomitantly used a different approach to obtain the corresponding extension principle with finite blueshift (Doppler exponent) for the uncharged Einstein-scalar field system. Hence, we decided to put the current paper as an appendix to An-Tan [2], thereby completing the proof of the weak cosmic censorship for the Einstein-Maxwell-charged scalar field system. At the same time, we are working with Cicortas and his collaborators to write a new preprint containing both approaches for the Einstein-scalar field system.

We present a new quantitative, scale-critical $C^1$ extension principle for the spherically symmetric Einstein–scalar field system. We derive estimates for solutions in the causal past of a possibly singular endpoint $\mathcal{O}$ of the center. Assume that there is no trapped surface in this region and assume that the Doppler exponent (blue shift) along each incoming null cone is uniformly bounded. We then prove that the solution can extend regularly across $\mathcal{O}$ provided the mass ratio $\mu=2m/r$ satisfies $\sup\mu<\frac38$. This improves the small-mass-ratio condition in Christodoulou's extension principle by an explicit threshold of order one. Our proof has three new ingredients. First, we derive several new uniform scale-critical pointwise estimates for the scalar field and its derivatives. Second, we obtain refined estimates in an exterior self-similar domain within the triangular causal past region. Third, we establish a family of higher-order Morawetz-type identities and identify the ranges of $\mu$, with which there are coercive spacetime estimates. The unweighted identity yields the threshold $\mu\leq\frac14$, while employing extra self-similar weight raises the threshold to $\mu<\frac38$ and provides a weighted control, which implies that the solution's BV norm along an outgoing null cone is small, so that Christodoulou's theory of bounded variation solutions gives the extension.

In this short note, using OpenAI's resolution of the Solomon–Yau conjecture, we extend our joint work with Pedro Gaspar to all dimensions $n\geq2$. We construct eternal Allen–Cahn flows on $S^{n+1}$ connecting approximations of any prescribed least-area Clifford hypersurface and equatorial sphere. We also construct infinitely many eternal integral Brakke flows from least-area Clifford hypersurfaces to equatorial spheres.

Let $E\to B$ be a smooth $S^n$ bundle with oriented Riemannian total space $(E,g)$, where $n\ge2$. We prove K-area$(E,g)\le4\pi n(n-1)/\sigma$ whenever every fibre has intrinsic scalar curvature at least $\sigma>0$, independently of the dimension of $B$. In particular, unit round fibres give K-area$(E,g)\le4\pi$. For arbitrary metrics on two-sphere fibres, K-area is bounded by the supremum of their areas. We also obtain stronger bounds from fibrewise scalar spectra.

We investigate the notion of holomorphic $p$-contact structure, that we introduced recently in the scalar case with H. Kasuya and L. Ugarte and in the line-bundle case with K. Broder, in the context of a holomorphic family of compact complex manifolds equipped with a holomorphic family of holomorphic line bundles. Firstly, we show that the existence of a holomorphic $p$-contact structure is deformation-open when a non-jumping assumption is made on the $\bar\partial$-cohomology of bidegree $(p,\,0)$. This ought to become a source of further examples of holomorphic $p$-contact manifolds. Secondly, we obtain a Berndtsson-type (semi-)positivity result for the holomorphic vector bundle over the base induced, as a direct image, by the line bundle on the total space that supports the p-contact structures. Thirdly, we introduce a moduli space of relative holomorphic p-contact structures and we show that it is Hausdorff and open in a certain quotient topological space.

In this article, we provide a complete answer to one of the most classical conjectures in differential geometry (of foliations). More precisely, we present a complete positive answer to the classical conjecture, due to Meeks III, Pérez, and Ros (2008), that states that every foliation on $\mathbb{R}^{n+1}$ by (possibly varying) CMC hypersurfaces is a foliation by minimal hypersurfaces. Moreover, if $n\leq 6$, such a CMC foliation must consist of parallel hyperplanes.

We address a question of Canary–Taylor and Canary–Minsky–Taylor for sufficiently symmetric compact hyperbolic $3$-orbifolds with totally geodesic boundary. We give sufficient conditions on the symmetries of such an orbifold to show that, among the complete convex cocompact structures on the interior, the structure with Fuchsian ends is a critical point (respectively, a strict local minimum) for the critical exponent. We also provide an infinite family of non-rigid examples fulfilling these conditions.

Our previous work established the mapping cone Chern character on the even-degree side, using the powers of the mapping cone curvature associated with the given mapping cone connection. In this paper, we carry out the odd-degree version using a trivial mapping cone connection and an automorphism on a trivial complex vector bundle. This version is based on the related odd-degree transgression.

We prove stability of geodesic spheres under volume-preserving mean curvature flow in the simply connected space forms under genuinely first-order geometric assumptions. The initial hypersurface is required to lie in a sufficiently thin annulus around a geodesic sphere and to have outward normal making a small angle with the radial direction. No convexity, mean convexity, initial curvature bound, or Hölder control of the normal is assumed. The main ingredient is a quantitative Alexandrov reflection argument that converts these initial assumptions into a strong reflection property preserved by the flow. The resulting radial and angular control allows us to derive estimates for the mean curvature and the full second fundamental form, and hence long-time existence, eventual convexity, and smooth exponential convergence to a geodesic sphere. In hyperbolic space, the eventual convexity can be strengthened to horospherical convexity. The reflection threshold also yields a quantitative estimate for the displacement of the limiting centre. For Lipschitz initial hypersurfaces, we develop a separate positive-time curvature estimate that is uniform along smooth approximations. This produces an instantaneously smooth evolution with the same eventual convexity and smooth exponential convergence.

We classify, up to equivalence, all left-invariant complex structures on the three-dimensional Euclidean rigid motion group $SE(3).$ Our approach is based on the correspondence between complex structures on a real Lie algebra $\mathfrak{e}(3)$ of the motion group and complex Lie subalgebras of the complexification of $\mathfrak{e}(3).$ The resulting structures form a one-parameter family $J_\phi.$ We further establish a connection with the classical Calabi-Eckmann complex structures on $S^3\times S^3.$ More precisely, we construct a one-parameter deformation of Lie brackets connecting $\mathfrak{so}(3)\oplus \mathfrak{so}(3)$ to $\mathfrak{e}(3),$ for which each endomorphism $J_\phi$ remains integrable. Consequently, every left-invariant complex structure on $SE(3)$ arises as a contraction limit of Calabi-Eckmann structures. The same family of endomorphisms also defines complex structures on $\mathfrak{so}(3,1).$

We establish new restrictions on the virtual cohomological dimension of word hyperbolic groups admitting Anosov representations into $\operatorname{PO}(p, q)$ ($p > q \geq 2$) or the exceptional Lie group $\operatorname{G}_2'$. Our main result states that, if $P_k$ is the parabolic subgroup of $\operatorname{PO}(p, q)$ stabilizing an isotropic $k$-plane in $\mathbb{R}^{p, q}$, a group admitting a $\{P_1,P_q\}$-Anosov representation into $\operatorname{PO}(p, q)$ has virtual cohomological dimension strictly less than $p$. As a result, a $\mathbf{H}^{p, q-1}$-convex cocompact subgroup of $\operatorname{PO}(p, q)$ cannot act properly discontinuously and cocompactly on $\mathbf{H}^{p, q-1}$. The main theorem also implies that a group admitting a $P_{\alpha}$-Anosov representation into the exceptional Lie group $\operatorname{G}_2'$ has virtual cohomological dimension strictly less than $4$, where $\alpha$ is a short restricted root of $\operatorname{G}_2'$. As an application, we prove that the homogeneous space $\operatorname{G}_2'/ \operatorname{SU}(2, 1)$ does not admit any compact quotient.

A Lagrangian submanifold in an almost Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. It is called $\epsilon$-positive if the ratio of the restrictions of the imaginary and real parts of the holomorphic volume form is bounded by $\frac{1}{\epsilon}.$ An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We prove that the Riemannian metric restricted to the subset of $\epsilon$-positive Lagrangians induces a path metric when the relative Kähler class of the ambient Calabi-Yau manifold lies on a line of rational slope. The proof relies on the construction of a sequence of quantization maps from a circle bundle over the space of positive Lagrangians to the dual of the space of holomorphic sections of powers of a prequantum line bundle. The rationality condition for the relative Kähler class allows us to reduce to the case of Bohr-Sommerfeld Lagrangians. Building on ideas of Tian and Nirenberg-Wells, Hörmander estimates are used to show that functions on a Bohr-Sommerfeld Lagrangian can be approximated by holomorphic sections of a sufficiently high power of the prequantum line bundle. Consequently, the quantization maps eventually separate fibers of the circle bundle.

Generalized complex structures give rise to generalized Dolbeault, Bott–Chern and Aeppli cohomologies. We study these cohomologies for invariant generalized complex structures on almost abelian Lie algebras $\mathfrak g_A=\mathbb R e_0\ltimes_A\mathfrak h$, where $\mathfrak h$ is an abelian ideal of codimension one, and on almost abelian solvmanifolds $\Gamma\backslash G_A$. We first show that the invariant generalized Dolbeault cohomology can be identified with the Chevalley–Eilenberg cohomology of the complex Lie algebra $\mathcal L$, the $+{\mathbf i}$-eigenspace of the structure, with coefficients in a character determined by the canonical line. We prove that $\mathcal L$ is itself an almost abelian Lie algebra; thus the cohomology reduces to the kernel and cokernel of an explicit derivation. Assuming that $A$ is diagonalizable over $\mathbb R$, we obtain a combinatorial formula for the cohomology dimensions and compare with compact quotients. Other comparison results are discussed, noting that the cohomology of invariant forms always injects into that of $\Gamma\backslash G_A$.

In this paper, we solve the direct monodromy problem for the Borel-summed $0$-parameter solution of Painlevé~I. Using the auxiliary time equation, we first construct formal WKB wave matrices whose amplitudes are regular at the double turning point. Rigorous Borel summation then yields exact WKB matrices in each Stokes region. Uniform estimates on moving action strips provide a common exact wave matrix in a full neighbourhood of the double point, proving that all four incident jumps are identities for consistent column normalisations. We formulate the resulting normalised Riemann–Hilbert problem and prove its existence and uniqueness on the Borel domains of the construction. Finally, after Painlevé scaling, the five Borel-summed solutions coincide with the five rotated tritronquée solutions of Painlevé~I.

We show that Payne's conjecture on the comparison between Dirichlet Laplacian eigenvalues and buckling eigenvalues does not hold on flat and hyperbolic cylindrical surfaces and on (round) hemispheres.

Let $\Gamma$ be a connected metric graph of first Betti number $b\ge 2$, and let $\Gamma_b$ be a trivalent graph of the same Betti number with all edges of unit length. We show that if $\text{length}(\Gamma) \le \lambda\, \text{length}(\Gamma_b)$, then the universal cover of $\Gamma$ contains a vertex around which, at every radius $R\ge 0$, the ball is at least as long as the ball of radius $R$ in the trivalent tree whose edges all have length $3\lambda$. In particular, for $\lambda\le\frac13$, this ball is at least as long as the ball of the same radius in the universal cover of $\Gamma_b$, with no multiplicative loss. This improves an earlier theorem of the author, which gives the weaker lower bound $(1-3\lambda)$ times the ball in the universal cover of $\Gamma_b$, and only for $\lambda\le\frac16$. The proof rests on an elementary comparison principle. Among trees whose vertices have degree at least three and whose edges have length at most $c$, the trivalent tree with all edges of length $c$ has the smallest balls at every radius. As an application, we double the exponential growth rate in the author's earlier uniform lower bound for the area of balls in the universal cover of small-area surfaces of hyperbolic type, from $\ln 2$ to $2\ln 2$, beyond the hyperbolic rate. We also obtain explicit constants in the positive answer to Guth's question for surfaces. Finally, we conjecture the sharp, all-scale version of the minimal volume entropy inequality of Kapovich–Nagnibeda and Lim, and show that it would be optimal.

The horizontal mean curvature flow is a geometric evolution associated with the horizontal perimeter in sub-Riemannian spaces. In this paper, we introduce a minimizing movement scheme for this flow in the first Heisenberg group $\mathbb{H}$, and prove the existence of a flat flow which is locally $1/2$-Hölder continuous in time, with respect to the $L^1$-distance. The construction applies to arbitrary bounded sets of finite horizontal perimeter and does not require smoothness or symmetry of the initial datum.

We fix a spacelike two-dimensional left-invariant distribution and its associated almost contact structure on a three-dimensional Lorentzian Lie group, and investigate the star-Ricci contraction of the curvature of the Bott connection. We compute the complete tensor for seven standard bracket families and determine all real parameter solutions of the general algebraic equation and its self-adjoint subclass. Using the symmetric part of the tensor, we also define a Bott affine star-Ricci soliton equation and classify all left-invariant vector field solutions. Every affine solution is steady, while two non-unimodular families admit branches with a nonzero symmetrized tensor. The proofs provide complete component equations, treat all degenerate parameter branches, and establish sufficiency by explicit substitution. We also distinguish almost contact structures from contact metric structures.

We study unfoldings of foliated complex analytic structures defined by locally free subsheaves of cotangent sheaves. We prove theorem of existence for unfoldings of foliated complex analytic structures defined by locally free subsheaves of cotangent sheaves for one-parameter families. We prove theorem of completeness for unfoldings of foliated complex analytic structures defined by locally free subsheaves of cotangent sheaves. We also present unfoldings of foliated nonsingular varieties defined by locally free subsheaves of cotangent sheaves in the language of functors of Artin rings which is the algebraic version of unfoldings of foliated complex analytic structures. We identify first-order unfoldings and obstructions.

We give a parabolic proof of the Fu-Wang-Wu theorem for semilinear elliptic equations associated with generalized Gauduchon metrics. Fixing a Hermitian metric $\omega$, we introduce a flow $\partial_t v-\Delta v=\psi(\vert\nabla v\vert^2)+\langle B,dv\rangle-c(t)-f(x)$ for the conformal factor, whose stationary points solve the semilinear elliptic equation of Fu-Wang-Wu. Under the growth condition $\liminf_{t\to+\infty}\psi(t)/t^\mu>0$ for some $\mu>\tfrac{1}{2}$, we prove that the flow exists uniquely for all time and converges; the limit solves the elliptic equation and is unique up to addition of a constant. Our convergence argument does not rely on a gradient-flow structure or a monotone energy functional. In particular, this recovers, by a flow method rather than the continuity method, the existence and uniqueness theorem of Fu-Wang-Wu on $k$-th generalized Gauduchon metrics, providing a new parabolic approach to their elliptic theory.

Let $M$ be a rank-one symmetric space of compact or noncompact type and real dimension $m=kn$, where $k\in\{1,2,4,8\}$ is the real dimension of the underlying division algebra. Meng and Wang proved sharp isoperimetric inequalities for reciprocal sums of the first $p$ or $\ell$ nonzero Neumann eigenvalues, with indices strictly below $m$ in the nonconstant-curvature cases. We keep the whole $m$-dimensional transplanted first eigenspace of an equimeasurable geodesic ball and treat its mass and stiffness forms as a coupled generalized eigenvalue problem. For every bounded connected smooth domain in a noncompact rank-one symmetricspace we prove the full sharp estimate \[ \sum_{j=1}^{m}\frac{1}{\mu_j(\Omega)} \ge \frac{m}{\mu_1(B)}, \] where $B$ is a geodesic ball with $|B|=|\Omega|$; equality characterizes geodesic balls. The Hopf-direction contribution is encoded by a positive, unital, trace-preserving projection on symmetric matrices, and the finite-dimensional closure is supplied by a positive-projection Jensen inequality. In compact type, under the localization hypothesis of Meng–Wang, the same full estimate is obtained whenever $\cot^2 R\ge k-1$, where $R$ is the radius of the comparison ball, as well as in the isotropic endpoint cases and for Hopf-balanced angular second moment. Consequently the complete $m$-term inequality holds for all complex and quaternionic projective spaces under the original localization assumption, and for the Cayley plane when $R\le \arctan(1/\sqrt7)$. In the remaining admissible Cayley interval we exhibit a finite-dimensional obstruction showing that the affine tangent-matrix bounds alone cannot yield the full trace estimate. This obstruction concerns the proof mechanism and is not a counterexample to the unresolved spectral inequality.

We determine the large-genus order of the Weil-Petersson diameter of $\mathcal{M}_{g,n}$ for each fixed number of punctures: namely, $\mathrm{diam}(\mathcal{M}_{g,n})\asymp\sqrt g\log g$. The main new ingredient is an integrated Ricci estimate for geodesics that minimize in moduli space and have thick endpoints, with no systole bound along their interiors.

This short article is devoted to the study of the positive solution of Yamabe-type semi-linear equation with prescribed coefficient on general closed manifolds. Our main result is to provide some sufficient conditions for the existence of the positive solutions and give some examples to show that those conditions are sharp. An application to scalar-flat manifold settles a long standing question in prescribing scalar curvature problem in negative form.

We prove the fence conjecture of Dumas and Wolf: for a convex projective polygon with at least five vertices, every zero of the Pick differential lies in the interior of the region obtained by cutting off the triangles at consecutive triples of vertices. The proof gives a stronger estimate for arbitrary properly convex domains. In affine-sphere coordinates normalized at a Pick zero, the domain contains the closed disk of radius $2^{-1/4}$ and its closure lies in the open disk of radius $2^{1/4}$, unless it is a rotated square. The proof compares the canonical Monge-Ampère potential of the domain with that of a normalized square. Their third-order contact gives alternating local sign sectors. A planar separation argument and the maximum principle show that the negative sectors belong to four distinct components, each meeting a different side of the square. Rotation gives the inner disk bound, and conormal duality gives the outer bound. Both constants are optimal for the class of properly convex domains normalized at a Pick zero. The argument requires neither boundary smoothness nor any restriction on the multiplicity of the Pick zero.