Papers

Let \(\mathbf{E}\to M\) be a finite-rank Riemannian vector bundle over a compact Riemannian manifold with nonempty smooth boundary, equipped with a compatible connection. We develop an intrinsic trace theory for Sobolev sections defined through weak covariant derivatives. For every integer \(m\ge1\) and every \(1\le p<\infty\), we construct continuous trace operators for the covariant derivatives up to order \(m-1\) and characterize the Sobolev space with homogeneous Dirichlet boundary conditions as the kernel of the corresponding trace operator. As an application, we establish the existence of ground state solutions for a class of quasilinear elliptic equations on Riemannian vector bundles with generalized \((p,q)\)-growth. The proposed framework unifies several important geometric operators, including the Bochner \(p\)-Laplacian, the \((p,q)\)-Bochner Laplacian, and prescribed mean curvature-type operators.

In 2002, Montiel and Urbano conjectured that the Clifford torus in $\mathbb{C}P^2$ minimizes the Willmore-type functional $ \mathcal{W}^-=\int_{T^2}(2+|H|^2)\,dA,$ either among all tori or among all Lagrangian tori. In this paper, we confirm this conjecture in the Lagrangian setting and disprove it in the general setting. We establish that every oriented closed Lagrangian surface of genus $g\geq 1$ in $\mathbb{C}P^2$ has $\mathcal{W}^-$-energy no less than that of the Clifford torus. Moreover, we construct non-Lagrangian deformations of the Clifford torus along which $\mathcal{W}^-$ strictly decreases.

Let $(M^3,g)$ be a complete, connected, noncompact Riemannian three-manifold without boundary and with nonnegative sectional curvature. We prove that its scalar-curvature integral over geodesic balls, divided by the radius, has a limit. In the one-ended case, \[ \lim_{r\to\infty}\frac1r\int_{B_p(r)}\operatorname{Scal}\,d V =8\pi\bigl(\chi(M)-V_M\bigr)\le8\pi(1-V_M), \] where $V_M$ is the asymptotic volume ratio. In the one-ended case, we show that $\chi(M)\in\{0,1\}$. Thus positive asymptotic volume ratio gives the value $8\pi(1-V_M)$, while in the collapsed case the value is determined by the topology of the end. No pole or scalar-curvature bound is assumed. The proof combines separate smooth approximations of the Busemann and distance functions, integrable negative curvature errors, and a determinant estimate on the level surfaces. An averaged boundary estimate then gives convergence of the integrated extrinsic curvature. In the two-ended case, the splitting theorem gives the exact limit $8\pi\chi(N)$ for the compact surface factor $N$. The one-ended upper bound is attained for every prescribed asymptotic volume ratio in $[0,1]$, and the two-ended bound is also sharp.

We classify compact Kähler surfaces with semipositive anticanonical bundle. In particular, we resolve a question of Demailly–Peternell–Schneider: any blowup of $\mathbb{P}^2$ at 9 points on a smooth cubic has semipositive anticanonical bundle.

We show that a great circle in $\mathbb{S}^3$ bounds embedded non-orientable minimal surfaces of unbounded genera and all possible non-zero Euler numbers. This answers a question of R. Hardt and H. Rosenberg (1990). It also gives the first example of a real analytic Jordan curve bounding infinitely many embedded minimal surfaces of distinct topological types. Such surfaces may arise as the links of boundary singularities of non-orientable minimal hypersurfaces in $\mathbb{R}^4$.

We prove that every closed cooriented hypersurface immersion in $\R^{n+1}$, $n\ge 2$, with positive definite first Newton transformation bounds a compact immersed manifold with the prescribed boundary map and outward coorientation. Combined with the rigidity results of Ros and Pinkall, this filling theorem implies that every closed connected hypersurface immersed in $\R^{n+1}$ with constant scalar curvature is a round sphere.

For every integer $n\ge3$, we construct a smooth projective manifold $X$ of complex dimension $n$ whose canonical bundle is not pseudoeffective, or equivalently, which is uniruled, but every Kähler metric has negative total scalar curvature. In particular, $X$ admits no Kähler metric of positive scalar curvature, while it admits a Riemannian metric of positive scalar curvature. Thus the equivalence between uniruledness and the existence of a Kähler metric with positive scalar curvature holds in complex dimensions one and two, but fails in higher dimensions.

We prove the sharp two-disk comparison conjectured by Langford and Laugesen for the second positive Neumann eigenvalue under an upper Gaussian-curvature bound, and strengthen it to a shifted reciprocal inequality. Let $\Omega\subset\mathbb{C}$ be bounded, simply connected and Lipschitz, with $g=\omega|dz|^2$, where $\omega$ is continuous and positive on $\overline{\Omega}$ and $\Delta\log\omega+2K\omega\ge0$ in distributions. Write $A=|\Omega|_g$ and enumerate the spectrum by $0=\lambda_0<\lambda_1\le\lambda_2\le\cdots$. If $KA<4\pi$ when $K>0$, then $\frac{1}{\lambda_2}+\frac{1}{\lambda_3}>\frac{2}{\lambda_1(D_K(A/2))}$ and $\lambda_2<\lambda_1(D_K(A/2))$, where $D_K(A/2)$ is the constant-curvature disk of area $A/2$. The same conclusions hold for disk-type Lipschitz domains in smooth surfaces. In the subcritical regime, both bounds are sharp, as witnessed by an extremizing sequence of smooth connected domains of exact area and curvature whose Neumann eigenvalues converge, at every fixed index, to those of two disjoint equal model disks. At $KA=4\pi$ we obtain the sharp non-strict bounds $\lambda_2\le2K$ and $\lambda_2^{-1}+\lambda_3^{-1}\ge K^{-1}$. The proof combines two-pole Green coordinates, positive-kernel comparison, and simultaneous centering. We also establish a global quadratic deficit for the model area split.

We prove a priori interior $L^2$ bounds for the second fundamental form of area-minimizing currents modulo $2$ in every dimension $n\geq 2$ and arbitrary codimension. The bounds depend only on the dimension, codimension, and an upper bound for the mass. As a corollary, we show that area-minimizing surfaces modulo $2$ with bounded area lie in finitely many diffeomorphism classes. This answers a question of \cite{Liu2020WeakCounterexamples}.

In this paper, we establish a general volume estimate for complete Riemannian manifolds under suitable differential inequalities involving a proper function and a symmetric tensor, without imposing curvature assumptions. As an application, we prove that a class of orientable hypersurfaces properly immersed in Euclidean space has at most polynomial volume growth and finite weighted volume. This class includes, in particular, certain self-shrinkers of the $Q_k$-flow and properly immersed minimal hypersurfaces for which the normal component of the position vector is bounded.

In this paper, we prove the optimal diameter estimate for positive Yamabe metrics on a connected, closed manifold. It is a scalar curvature version of both classical Myers' diameter estimate and Cheng's maximal diameter theorem on Ricci curvature.

We construct complete ALH hyperkähler metrics on complements of anticanonical elliptic curves with zero self-intersection in smooth rational surfaces. This answers the existence question of Filip–Tosatti, related to the picture proposed by Haskins–Hein–Nordström. Conversely, we prove that every hyperkähler ALH gravitational instanton admits, for each induced complex structure, a compactification by an anticanonical elliptic curve in a rational surface.

A Kähler structure on a real Lie algebra $\mathfrak{g}$ is a pair $(\omega, J)$ of a 2-cocycle and a complex structure on $\mathfrak{g}$ which are compatible in the sense that $J$ is skew-symmetric with respect to $\omega$. This paper investigates Kähler structures on Lie algebras, with a particular focus on the nilpotent case. We introduce an algebraic framework for the Kählerian reduction and oxidation of Lie algebras by one-dimensional and two-dimensional ideals. Our main structural result demonstrates that every indecomposable, quasi-nilpotent Kählerian nilpotent Lie algebra of dimension greater than four can be inductively reconstructed through a finite sequence of Kählerian central oxidations by planes. This reduction sequence originates from a base algebra that is either $\{0\}$, $\mathbb{R}^2$, or one admitting a strongly non-nilpotent complex structure. Moreover, if the complex structure is nilpotent, all intermediate reductions also have a nilpotent complex structure.

A metric space is called all-set-homogeneous if every isometry between two of its subsets extends to an isometry of the whole space. We classify all-set-homogeneous geodesic spaces: besides the classical examples, they include the universal metric trees of finite valence. We also prove that every complete all-set-homogeneous length space is geodesic, and hence the same classification holds in this setting.

We provide the first compact examples of type-changing generalized Calabi–Yau manifolds using Fei's construction of non-Kähler Calabi–Yau manifolds on branched covers of twistor spaces \cite{Fei16}. The structure, which is generically of type 1 and changes to type 3, is constructed on the product of a hyperelliptic surface of genus three and a four-torus, $\Sigma_3 \times \mathbb{T}^4$, with starting point a non-Kähler Calabi–Yau structure.

We establish lower bounds for the critical CR GJMS operator on closed pseudo-Hermitian manifolds. In dimension $2n+1$, assuming embeddability, we obtain a lower bound involving the Heisenberg Sobolev norm of order $n+1$, modulo an $L^{2}$ term, on the orthogonal complement of the kernel. In dimension three, we prove that the CR Paneitz operator is bounded below without assuming embeddability. The proofs combine the Heisenberg calculus with a square-root construction based on the joint functional calculus of the sub-Laplacian and the Reeb vector field on the Heisenberg group. As an application, we show that the infinitely many negative eigenvalues of the CR Paneitz operator on the Rossi sphere have zero as their only accumulation point. Consequently, this operator does not have closed range.

For any complete noncompact manifolds $(M^n, g)$ of nonnegative sectional curvature, with $3\le n\le6$, we obtain $\displaystyle \lim_{r\to\infty}r^{2-n}\int_{B_p(r)}\Sc_g\,dV_g\le8\pi\omega_{n-2}(1-\AVR(M^n,g))$ and the corresponding rigidity. The latter bound and its equality classification hold in $(M^n, g)$ with positive-dimensional souls. With a pole, $\displaystyle \lim_{r\to\infty}r^{2-n}\int_{B_p(r)}\Sc_g\,dV_g\le4\pi\omega_{n-2}(1-\AVR(M^n,g))$ follows from an independent distance-sphere proof. We also prove the sharp total scalar-integral bound for closed manifolds whose c

For the semistable $J$-equation, we prove that the numerical $J$-null locus coincides with the ambient $C^2$-singular locus of Murakami's weak solution. Consequently, this singular locus is a proper analytic subset with finitely many irreducible components, and the weak solution is locally smooth outside the $J$-null locus. The proof relies on two analytic ingredients. First, we establish a regularization theorem for singular $J$-subsolutions, showing that Demailly's global regularization preserves quantitative strict cone conditions. Thus, singular strict subsolutions with prescribed analytic poles can be replaced by smooth strict subsolutions away from their pole sets. Second, we derive relative a priori estimates adapted to these subsolutions: a relative $L^\infty$-estimate from determinant control and a weighted second order estimate yielding uniform $C^2$-bounds on compact subsets of the regular locus.

Based on an inequality on the upper bound of the sum of squared norms of Lie bracket of symmetric matrices, we establish a rigidity theorem for compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices, which are the critical values of the squared norms of the second fundamental form on all normal directions. This conclusion is optimal for all dimensions without the restriction of the codimension, giving a new characterization for generalized Clifford tori and Veronese manifolds.

We study weakly Einstein and weakly $\eta$-Einstein structures on almost contact metric three-manifolds. We first obtain a pointwise structural description of the Ricci operator $Q$ on a weakly $\eta$-Einstein almost contact metric three-manifold. More precisely, at each point, either $Q$ has the $\eta$-Einstein form, $Q\xi=0$, or the metric is weakly Einstein. We then prove that if a contact metric three-manifold satisfies $Q\xi=0$, then either $Q=0$ or $\operatorname{rank}Q=1$. Hence, such a metric is either flat or weakly Einstein but not Einstein. We also construct examples showing that the contact metric assumption is essential: there exist almost contact metric three-manifolds satisfying $Q\xi=0$ whose Ricci operators have rank two. Finally, we classify simply connected homogeneous contact metric three-manifolds that are weakly Einstein.

Equivariant convolutional neural networks are usually built from a group acting globally on the space of signals. This hypothesis is inappropriate for many bounded or stratified domains: an ambient rigid motion may be admissible only on part of the domain, and the boundary introduces geometric types that are invisible to a transitive group action. We develop a theory of groupoid-equivariant neural networks in which the symmetry datum consists of a groupoid, a selected pseudogroup of local bisections, a measure, and input and output representation bundles. For integral channels on the object space, we prove a bisection-equivariant kernel theorem: equivariance is equivalent to a transport constraint on the two-point kernel, and its solutions are classified by one joint-stabilizer intertwiner on each orbit of pairs. As a case study we apply the theory to bounded planar domains. The resulting architecture is implemented through offline nullspace bases and sparse gather–transform–scatter operations. A Poisson–Dirichlet kernel study is used separately to assess boundary-aware inductive bias; the exact inverse is shown to preserve the global symmetries of the rectangle but not general proper local bisections. The numerical results show that the proposed architectures provide significant advantages when symmetries cannot be globally implemented by group actions and provide an accuracy improvement of at least one order of magnitude with respect to the models tested.

We introduce a Green–potential heat flow for maps from a closed Riemannian surface. Inspired by Parker–Wolfson renormalization~\cite{ParkerWolfson1993,Parker1996}, the conformal factor is reconstructed at each time from an energy-based source determined by the current map. The resulting parabolic–elliptic system incorporates renormalization into the evolution while leaving the Dirichlet energy functional and its harmonic critical points unchanged. We establish local well-posedness for $H^3$ initial data and a continuation criterion in terms of energy concentration. For sufficiently large coupling, we exclude finite-time bubbling and obtain global solutions. At infinite time, the energy identity provides only weighted $L^2$ control of the tension field. Localizing to regular regions of concentration charts yields harmonic profiles along subsequences of prescribed times. We represent these profiles in the flow's packet charts, organize them into a finite bubble tree, and decompose the limiting energy into harmonic component energies and neck energies. A critically normalized radial example illustrates how the modified flow can prevent finite-time blow-up and produce harmonic bubbles with no-neck property.

We study the structure of compact Vaisman and l.c.K. manifolds with nonnegative bisectional curvature in this paper. Our first main result (cf. Theorem 1.3) is a uniformization theorem for compact Vaisman manifolds, extending Mok's uniformization theorem in the Kähler setting (cf. [Mok88]). Our second result (cf. Theorem 1.6) gives a structural trichotomy for compact l.c.K. manifolds with nonnegative Chern bisectional curvature: the underlying complex manifold admits a Kähler metric, admits a Vaisman metric, or its universal cover is conformally equivalent to the product of an incomplete Kähler manifold and a positive-dimensional complex Euclidean space.

We construct the moduli space of marked polystable $\mathrm{SL}_r(\mathbb C)$-Higgs bundles via analytic methods. The resulting space is normal and Hausdorff, and its local models are products of a Teichmüller neighbourhood with a quotient of a quadratic cone. These models preserve the curve parameter and identify the stabilizers. We prove agreement with the complex and unitary gauge-quotient topologies and establish the coarse moduli property for semistable families over reduced analytic bases. The mapping class group and Higgs scaling act holomorphically. Besides, on the stable locus, the moduli spaces that we consider agree with the joint moduli constructed by Collier, Toulisse and Wentworth.

We study the relation between canonical positivity, rigidity of a prescribed Hermitian metric, and the curvature not detected by Chern holomorphic sectional curvature. A compact Brody-hyperbolic complex manifold admitting an independent smooth plurinegative Hermitian metric is projective with ample canonical bundle. In Fujiki's class, nonpositive constant Chern holomorphic sectional curvature forces the original metric to be Kähler. On compact complex threefolds, without an auxiliary metric or a Fujiki-class assumption, a nonzero constant forces Kählerness. We also give an alternative proof of Chen–Zheng's Chern-flatness theorem for zero-HSC threefolds, by proving local rank-two differential compatibility and localizing their curvature bound under $\partial\eta=0$. In contrast, for every complex dimension at least seven we give compact balanced zero-HSC metrics with nonzero Chern curvature. Their octonionic curvature formula also implies the vanishing of all four Chern–Ricci contractions, and an explicit invariant volume form is Chern parallel. Finally, an auxiliary semipositive plurinegative $(1,1)$-form, positive definite at some point, restores flat Kählerness under zero HSC and vanishing first Chern–Ricci tensor. A mixed Ricci integral criterion explains the role of this last curvature hypothesis.

Let $g_t$ be a smooth family of Riemannian metrics on an oriented manifold of even dimension $n\geq4$, and let $D_t=d+\delta_{g_t}+\Psi_t$ on the fixed exterior bundle, with $\Psi_t$ self-adjoint. We evaluate a scalar-weighted residue functional obtained from the operator expression in Hawkins' Hamiltonian action using the coefficient derivative $\partial_tD_t$. The velocity contribution depends only on $\partial_tg_t$. A direct symbol calculation reduces the lapse correction to a divergence and a Dirichlet term; for a nonconstant test function, integration by parts retains a mixed gradient term. For compact manifolds with a warped collar, we specify compatible factorizations of the weighted KKW term and of the commutator correction. Their boundary residues are computed explicitly. The collar-derivative and perturbation contributions cancel in the weighted KKW factorization, and the remaining boundary terms involve normal derivatives of the scalar weights. The closed formula is recovered when the boundary is empty.

We study the supercritical LYZ equation on compact Kähler manifolds at the boundary of stability. Under numerical semistability, we prove a quantitative positivity estimate on the modified nef cone. It implies that the destabilizing prime divisors form a finite exceptional family. Assuming the existence of a smooth semisubsolution, we prove that the associated intersection form is negative definite on the span of classes of these divisors. Using this rigidity, we then construct a logarithmic singular subsolution along their union and obtain a bounded Bedford–Taylor solution which is smooth on its complement.

We prove a Positive Mass Theorem for $C^0$-asymptotically flat Riemannian metrics with nonnegative scalar curvature in a weak sense that are sufficiently uniformly close to Euclidean space. More precisely, we show that a $C^0$-asymptotically flat Riemannian metric that is a $C^0$ perturbation of Euclidean space with nonnegative scalar curvature in the sense of Ricci flow has nonnegative mass, where the mass is given by a $C^0$ analog of the classical ADM mass previously introduced by the author.

We study nonlinear mean-curvature spectral equations for hypersurfaces in Euclidean, spherical and hyperbolic space. For an admissible analytic response $F$ and ambient sectional curvature $c$, we prove minimality when the spectral parameter satisfies $\sigma\ge nc$, and strict mean-curvature bounds at gradient points when $\sigma<nc$. The class includes every power $F(t)=t^{p-1}$, $p>1$, and positive mixtures satisfying an explicit width condition. In particular, the result proves Chen's conjecture in Euclidean codimension one and biharmonic minimality in constant negative curvature, without completeness or holonomicity. A common real-analytic continuation and focal-trace argument treats all three geometries. Explicit umbilical models show that the unconditional spectral minimality range is sharp. Independently, we prove quantitative local rigidity, modulo ambient isometries, of compact nonminimal geodesic spheres in all three geometries and proper equal-radius Clifford hypersurfaces in spheres, for smooth laws with $F,F'>0$ near the model amplitude.

Following a program outlined by Bartnik, and also employing nonlinear potential techniques we developed in the last years, we study lightlike singularities of variational solutions to the prescribed Lorentzian mean curvature equation with possibly unbounded (rough) curvature. This corresponds to allowing for unbounded charges in the Born-Infeld model.