Papers

We prove the transcendental Morse inequality conjecture of Boucksom-Demailly-Păun-Peternell. Among the corollaries of this result, this proves orthogonality of divisorial Zariski decompositions, shows that the pseudoeffective and movable cones are dual, and establishes the differentiability of the volume function on the big cone, on all compact Kähler manifolds.

We study the set $\Gamma$ of graphs of closed, densely defined operators in a Hilbert space $H$, regarded as a subset of the Grassmann manifold $P(H\times H)$ of orthogonal projections in $H\times H$. We show that the subset $\Gamma^b$ of graphs of bounded operators is the open unit ball of $P(H\times H)$ centered at the graph of the zero operator $P_0$ (which projects onto $H\times\{0\}$). This ball is diffeomorphic to $B(H)$ via the map $T\mapsto P_T$ ($=$ the projection onto the graph ${Gr(T)}$ of $T$). We show that graphs of unbounded closed operators lie at the boundary of $\Gamma$. We also study the existence and characteristics of minimal geodesics of $P(H\times H)$ joining two graphs $Gr(A)$, $Gr(B)$. If $A,B$ are selfadjoint, such a geodesic always exists, and we construct explicitly a distinguished exponent using the five-space decomposition associated to the pair of subspaces $Gr(A)$, $Gr(B)$. An explicit low-dimensional example shows that the geodesic joining two graphs need not remain inside $\Gamma$, i.e., does not consist entirely of graphs. We also relate graphs of compact operators to the restricted Grassmannian, and study the problem of common complements for pairs $Gr(S)$, $Gr(T)$, giving positive results when one operator is bounded or under lower boundedness conditions.

We prove sphere theorems for closed connected immersed hypersurfaces of constant scalar curvature in Euclidean space. Combining Ros's rigidity argument with the filling theorem of Huisken and Sinestrari, we show that every two-convex such hypersurface is a round sphere. As a consequence, Yau's conjecture holds for closed connected immersed hypersurfaces in \(\mathbb R^4\). Using Gromov's filling theorem, we also obtain higher-dimensional rigidity results under a principal-curvature index condition and an explicit scalar-curvature pinching condition.

For every $n \geq 1$ there is a distinguished $\mathfrak{sl}_2(\mathbb{R})$ action on the Poincaré disk, arising as the infinitesimal isometries of a hyperbolic metric with conical singularity of order $n-1$ at the origin. For $n=1$ this is the standard infinitesimal Möbius action, and for $n>1$ these vector fields have singularities at the origin and are incomplete, preventing integration to a global Lie group action. However, they naturally define an action Lie algebroid $\mathcal{A}_n=\mathfrak{sl}_2(\mathbb{R})\ltimes \Dbarstar $ over the punctured disk. We construct an explicit Lie groupoid $\mathcal{G}_n$ integrating $\mathcal{A}_n$ and compare it to the \v Severa–Weinstein groupoid. Although $\mathcal{G}_n$ is not an action groupoid, its restriction to the boundary recovers an $n$-fold Möbius action on the boundary circle.

We discuss the dichotomy for fourth order Q curvature on Riemannian manifolds with positive Yamabe invariant and dimension at least 5.

We determine the long-time behavior of the canonical Ricci flow on a closed three-manifold of Thurston type $\mathbb{R}^3$, Nil, or Sol, starting from an arbitrary initial metric. In the Euclidean case, the metric $g(t)$ converges exponentially fast to a flat metric. The Gromov–Hausdorff limit of $(M,t^{-1}g(t))$ is a point in the Euclidean and Nil cases, and a metric circle or a compact interval of positive length in the Sol case. In all three cases, the blowdown flows lifted to the universal cover, $s^{-1}\widetilde g(s\tau)$, converge, in the pointed Cheeger–Hamilton sense, to an explicit homogeneous expanding Ricci soliton solution.

We regard the Chern torsion of a Hermitian manifold as a skew-symmetric complex-bilinear product on its holomorphic tangent bundle. For a compact connected Chern–Kähler-like Hermitian manifold, we prove that this product satisfies the Jacobi identity pointwise. If, in addition, the Chern holomorphic sectional curvature is strongly quasi-positive, we show that the resulting torsion Lie algebra is nilpotent and must be abelian. Consequently, the metric is Kähler. The underlying complex manifold is therefore projective and rationally connected. For general Hermitian manifolds, we construct a metric with positive real bisectional curvature on a Hopf surface which is neither simply connected nor rationally connected.

Let $M$ be a finite-volume hyperbolic $d$-manifold with $d\ge3$. We prove that the number of primitive nonsimple closed geodesics has exponential growth rate strictly smaller than $d-1$. Consequently, asymptotically almost every primitive closed geodesic in $M$ is simple. In contrast, we show that on an arithmetic hyperbolic manifold of type~I, the unit vectors tangent to nonsimple closed geodesics are dense in the unit tangent bundle.

We prove two sharp monotonicity formulae for the natural Green's function of a closed three-manifold with positive scalar curvature, analogous to the Munteanu-Wang theorem for non-negative scalar curvature. We also show that both results may be interpreted as the monotonicity of a suitably defined "enclosed mass" of the sublevel sets of Green's function in the corresponding "conformal" blow-up of the manifold.

We study two boundary cases of the stability condition for the $J$-equation. First, under the $J$-semistable condition, we show that the destabilizing prime divisors form an exceptional family in the sense of Boucksom, with a uniform numerical gap away from them. The related modified nef estimate can remove the $J$-big assumption in Liu's work~\cite{Liu2026Boundary}. Second, under the smooth boundary cone condition, we obtain a uniform global $C^0$ estimate for solutions of the approximating twisted $J$-equations. As a consequence, we construct a bounded-potential Bedford–Taylor solution of the $J$-equation, which is smooth outside the destabilizing prime divisors.

Inspired by interaction of gravitational waves, dark matters and dark energy, we study the Einstein scalar field equations with the nonzero cosmological constant for the Bondi-Sachs metrics. We provide the asymptotic expansions under the outgoing radiation condition. Unlike the case of the zero cosmological constant, the asymptotic expansions depend on four additional $\Lambda$-independent functions $B(u, \theta, \phi)$, $\tilde{X}(u, \theta, \phi)$, $\tilde{Y}(u, \theta, \phi)$ and $\tilde{I}(u)$. We show that, under suitable coordinate transformation, we can make $B(u, \theta, \phi)=0$. We prove the peeling property for the Einstein scalar field equations if $\tilde{I}=0$, and derive a loss formula of the Bondi energy-momentum for the nonzero cosmological constant. We remark that, for certain real data from gravitational waves, the loss formula is likely to indicate the loss property of the Bondi energy-momentum.

Cartan geometries are curved analogues of homogeneous spaces, and the correspondence space construction produces, from a Cartan geometry of one type, another of a different type over a larger base manifold. The unit tangent bundle of a Riemannian manifold is, in this language, a Cartan geometry of type $(\operatorname{Euc}(n),O(n-1))$ obtained in this way; we study Cartan geometries of this type in general, whether or not they arise as correspondence spaces. We show that every such geometry induces on its base manifold an almost contact metric structure, together with an orthogonal splitting of the tangent bundle and a compatible linear connection; conversely, these data determine the geometry, so that the correspondence is a bijection. When $n\ge3$, we single out a distinguished Cartan connection, which we call normal, by a condition on its torsion formulated in terms of the Spencer differential. This normalization is dictated by the correspondence space construction: the Cartan connection induced on the unit tangent bundle of a Riemannian manifold is normal.

We completely characterize the $n$-fold connected sums of $S^3\times S^4$ that admit Sasaki-Einstein structures by proving that such a structure exists if and only if $n$ is even.

In this paper, we consider energy identity for intrinsic stationary biharmonic maps into homogeneous spaces in supercritical dimensions, extending the corresponding result of Hornung-Moser [Anal. PDE. 2012] in critical dimension. The proof follows a similar strategy as that of Lin-Rivière [Duke Math. J. 2002]. A key ingredient is a conservation law for intrinsic biharmonic maps into homogeneous spaces, which allow us to derive higher regularity of the map.

Assuming the validity of the recently proposed “A compact complex threefold fibred by tori over the projective line, and the six-sphere”, we construct an exotic complex structure on $\mathbb P^{3}$, distinct from the point-blowup structures of Huckleberry, Kebekus, and Peternell. Then we perform an Atiyah flop to produce a complex structure on the standard smooth manifold $S^{2}\times S^{4}$.

This paper is a sequel to [9], where the first-named author constructed a family of approximate Hermitian Yang–Mills metrics $H_{0,\epsilon}$ on stable rank-two holomorphic vector bundles arising from double spectral covers over the product of two one-dimensional complex tori. We prove that these approximate metrics give an all-order, exponentially accurate asymptotic description of the exact Hermitian Yang–Mills metrics in the large Kähler limit. More precisely, the mean curvature of $H_{0,\epsilon}$ decays exponentially in every $C^k$-norm. Moreover, if $H_{1,\epsilon}$ denotes the exact Hermitian Yang–Mills metric and \[ H_\epsilon=H_{0,\epsilon}^{-1}H_{1,\epsilon}, \] then, after normalization, for every nonnegative integer $k$, there exist positive constants $C_k$ and $c_k$ such that \[ \|H_\epsilon- Id\|_{C^k}\leq C_k e^{-\frac{c_{k}}{\epsilon}}. \] The main analytic difficulty lies in the global $C^0$-comparison. Obtaining $C^0$-estimates for the coupled nonlinear Hermitian Yang–Mills system is intrinsically difficult; moreover, the equation controls only the contraction of the curvature, and hence only certain combinations of second derivatives, whereas one needs global control of the full matrix-valued metric.

We study a quadrature surface free boundary problem on a smooth compact finite-dimensional Riemannian manifold $(M,g)$. The problem is formulated as a shape optimization problem involving a Dirichlet problem for the Laplace–Beltrami operator, with a geometric condition on the free boundary. Under suitable uniform geometric assumptions on the admissible class, we establish its compactness and prove the stability of the corresponding Dirichlet problems, including strong convergence of the associated states in $H_0^1(M)$. We derive the first-order optimality condition for the associated shape functional. Finally, we establish a Riemannian comparison principle for the second fundamental forms and the mean curvatures of tangent boundaries at a contact point. These results provide a rigorous extension of the quadrature surface free boundary framework from the Euclidean setting to compact Riemannian manifolds.

Let $(M,g)$ be an asymptotically flat manifold such that its tangent bundle $TM$ is diffeomorphically asymptotically Euclidean. We prove a positive mass theorem and a positive mass rigidity theorem for a natural class of asymptotically flat metrics $\widetilde g$ on $TM$ exhibiting slow asymptotic decay. These metrics are constructed by interpolating, along the horizontal distribution of $TM$, between the Euclidean metric and the base metric $g$ near infinity. The main difficulty is that the induced metrics on $TM$ decay below the standard threshold for the ADM mass in dimension $2n$; nevertheless, we show that their asymptotic mass is well-defined in a generalized sense and is determined by geometric data on the underlying manifold $M$.

Partially totally real (PTR) submanifolds were introduced in Kähler geometry by distinguishing a totally real distribution and leaving its orthogonal complement unrestricted. In this paper we develop the corresponding framework for submanifolds of Sasakian manifolds tangent to the Reeb vector field. After separating the Reeb direction, we define the totally real and ambiguous distributions and show that anti-invariant, contact CR, hemi-slant and pointwise hemi-slant submanifolds occur as special cases of the Sasakian PTR framework. We establish the basic tangential and normal decompositions, study maximality and integrability, and derive the additional restrictions produced by the Reeb field. We then investigate the canonical morphisms \(P\) and \(F\), the geometry of the associated distributions, and PTR-submanifolds in Sasakian space forms. Explicit models are included to illustrate the principal structures and the differences from the Kähler case.

Let $(M^n,g)$ carry a conformal product structure whose adapted Weyl connection $D$ preserves orthogonal distributions of ranks $p,q\ge2$. We express the Faraday form $d\theta$ directly in terms of the Weyl curvature of $g$. If $S$ is the orthogonal involution determined by the splitting and \[ \mathcal K_W(X)=\sum_i W_{X,e_i}(Se_i) \] for any local orthonormal frame, then \[ (d\theta)^\sharp = \frac{n-2}{4(p-1)(q-1)}[\mathcal K_W,S]. \] No Ricci-curvature assumption is required. The same curvature calculation gives a companion formula for the symmetric part of $\nabla\theta$, \[ S\Theta^s-\Theta^sS+\theta^\sharp\wedge S\theta^\sharp = \frac{p-q}{n-2}(d\theta)^\sharp +\frac1{n-2}[S,\Ric^\sharp], \qquad \Theta=\nabla\theta. \] When the Ricci tensor is block diagonal with respect to the product splitting, these formulas lead to explicit identities for the two components of the Lee form. In dimension four, the rank-$(2,2)$ splitting determines an ambi-Hermitian pair. The classical ambi-Hermitian curvature identities imply local closedness when the Ricci tensor is invariant under both complex structures; we also recover this conclusion from the mixed Weyl trace above. Finally, if $(M^4,g)$ is compact and Einstein, no specialness assumption is needed: $D=\nabla^g$. Hence the universal cover is the product of two simply connected surfaces with the same constant Gaussian curvature.

We give explicit formulas for the spectra of the intertwinors on the spinor bundle over $S^1 \times S^{n-1}$, $n$ even, with the standard Lorentzian metric $g=-g_{{}_{S^1}}+g_{{}_{S^{n-1}}}$. As a special case, we construct conformally covariant differential operators of all odd orders with the leading term a power of the Dirac operator.

This paper studies nontrivial Yamabe gradient solitons occurring as complete immersed hypersurfaces with constant scalar curvature in space forms $\left(\overline{M}^n(C), \overline{g}\right)$. For solitons with non-vanishing gradients, we establish a structural characterization of the soliton by analyzing a regular level set $\Sigma$ of the soliton function. In particular, imposing a suitable condition on the traceless part of the second fundamental form, $\Phi^{\Sigma}_{\overline{M}}$ of $\Sigma$ as a submanifold of the space form, we prove that the soliton either decomposes as a Riemannian product of $\mathbb{R}$ and a totally umbilical submanifold of the space form, or satisfies a sharp lower bound on $\sup\limits_\Sigma\left|\Phi^{\Sigma}_{\overline{M}}\right|$. Furthermore, we show that the lower bound is attained if, and only if, the soliton is isometric to a Riemannian product of $\mathbb{R}$ and a parallel submanifold of the space form.

We construct new ambient and Poincaré metrics for the Fefferman conformal manifold over $3$-dimensional CR manifolds, starting from the associated self-dual Einstein ACH metric. Unlike the classical ambient and Poincaré metrics arising from approximate solutions to the complex Monge-Ampère equation, these metrics satisfy Einstein-Maxwell-type equations to infinite order rather than the Ricci-flat or Einstein equations. Since these metrics are determined to infinite order without ambiguity, they enable us to construct CR invariant differential operators and local CR invariants involving arbitrarily high order derivatives of the Tanaka-Webster curvature and torsion. As an application, we obtain the ambient metric construction of the CR GJMS (Gover-Graham) operators of all orders in dimension $3$.

This article is a sequel to “A novel mathematical model of protein interactions from the perspective of electron delocalization (2025)”, where protein molecules are modelded as loops of triangles with no singular holes inside. Here we consider two questions: (1) How can we define the shape of a molecule? (i.e., how can we define the shape of a loop with no singular holes inside?), and (2) Is the given loop a molecule? (i.e., are the holes within a loop are singular?). These questions are answered by (1) giving a defining system of equations for the shape of a molecule using concepts from category theory, and (2) giving a equation that determines whether holes within a loop are singular using concepts from cohomology theory, respectively. No prior knowledge of the previous paper, category theory, or cohomology theory is required.

We prove a sharp nonlinear analogue of the Li-Yau inequality for positive solutions of the normalized parabolic $p$-Laplacian equation \[ u_t = \Delta u + (p-2)|\nabla u|^{-2}\nabla^2u(\nabla u,\nabla u),\ \ \ 1<p<\infty, \] on a closed Riemannian manifold with nonnegative Ricci curvature, the equation being interpreted in the viscosity sense on the critical set of the solution. We also derive a sharp global Harnack inequality from the gradient estimate. The constant is best possible. Equality holds identically for an explicit self-similar solution in $\Rn$, and sharpness on closed manifolds follows by taking the limit along the flat tori $\mathbb R^n/(L\mathbb Z)^n$ as $L \to \infty$. The proof rests on an exact Bochner-type identity for a family of uniformly parabolic approximating flows. We choose the first-order term separately from the second-order regularization so that the maximum principle yields the sharp constant, uniformly in the regularization parameter and without any assumption on the critical set of the solution. Using this Bochner identity with a modified Li-Yau functional, we extend the estimate to closed manifolds with $\operatorname{Ric} \ge -\kappa$.

This article investigates the global structure of maximal surfaces (space-like immersions with zero mean curvature) in Lorentz-Minkowski $3$-space $\mathbb{E}^3_1$, especially focusing on the interplay between genus, the number of singular components–loci, and simple ends. We construct complete maximal maps with arbitrarily many singular components for any genus $p\geq 0$ with simple ends.

In this paper we consider some general aspects of Euclidean conformal field theories on curved spaces of dimension $D\ge 3$. We first look at the OPE of scalar primary fields on conformally flat spaces. Extending the results of \cite{Konechny:2026bqg}, we give a general construction of the descendants' contributions to the OPE in the scalar channel. We then discuss CFTs on non-conformally flat spaces in the ambient space formalism. We investigate the short-distance behaviour of three-point functions on such spaces using the general ansatz proposed by Parisini, Skenderis, and Withers \cite{Parisini:2022wkb, Parisini:2023nbd}. We find that some additional corrections need to be added to the ansatz to ensure the existence of a local covariant OPE.

Let $M^n$ be a $C^2$ closed convex hypersurface in Euclidean space, and $\sigma_m$ be its $m$th mean curvature. We show that $M$ is symmetric with respect to a hyperplane orthogonal to a given direction $e$, if $\sigma_m(p)\leq\sigma_m(q)$ whenever $p-q$ is parallel to $e$. For convex hypersurfaces, this settles a conjecture of Li, extends the mean-curvature theorem of Li-Yan-Yao to all $\sigma_m$, and strengthens some earlier results of Li-Nirenberg by removing nondegeneracy assumptions. The proof is based on the theory of mixed volumes, specifically the rigidity of quermassintegrals under Steiner symmetrization.

A $\lambda$-self-expander $x: M^n\to \mathbb{R}^{n+1}$ is the solution of the isoperimetric problem of weight $e^{\frac{|x|^2}{4}}$. In this paper, we prove that the closed $\lambda$-self-expander is a round sphere centered at the origin under some curvature conditions. These curvature conditions mainly include the scalar curvature, the mean curvature, and the squared norm of the second fundamental form.

The reduction of mechanical presymplectic structures and its application to the reduction of time-dependent Hamiltonian systems were developed in a recent paper (Gutiérrez Sagredo et al, 2025). This approach overcomes some limitations of the original Albert reduction (Albert, 1989). In this paper we will describe the reduction of time-dependent Hamiltonian systems using the extended formalism. This process combines cotangent bundle reduction with reduction of presymplectic structures of corank 1 and 2. These results generalize previous work (Lacirasella et al, 2012), in the same direction. Several examples will be presented to illustrate the theory from this new perspective.