Papers

In this short note we prove a rigidity result for orientable minimal submanifolds in the round sphere of the lowest non-trivial Morse index. In particular, we show that the only closed and connected $(2m-2)$-dimensional orientable minimal submanifolds in $\mathbb{S}^{2m}$ of index less than or equal to $2m+1$ are the totally geodesic spheres.

We provide a complex-geometric approach to the Blaschke conjecture, i.e., that a manifold whose injectivity radius equals its diameter is isometric to a compact rank-one symmetric space (CROSS). In particular, we introduce the great quadric fibration, a complex analogue of the great sphere fibration. Requiring its total space to be a complex submanifold simultaneously explains many properties that a Blaschke manifold is expected to possess, including its Clifford structure, Hopf fibration, and diffeomorphism class. Furthermore, the great quadric bundle coincides with the variety of minimal rational tangents (VMRT) of the ambient Fano variety in the standard CROSS cases, and we explain this through a bend-and-break argument in the setting of the Blaschke conjecture under a suitable complexification assumption. Combining this with Hwang-Mok VMRT recognition, we prove the Blaschke conjecture for manifolds admitting an adapted complex structure on their entire tangent bundles.

We show that for closed negatively curved Riemannian manifolds whose length spectrum is exponentially separated, the length spectrum is a local isometry invariant. In particular, this holds for a dense set of negatively curved metrics. The main tool used in the proof is a finitary version of the Livshits theorem with an exponential tail.

M. E. Zieve conjectured that a candidate datum over the sphere is realizable whenever the gcd of the parts of every branch partition is one and it satisfies the lcm condition. We prove that this conjecture reduces to its three-branch-point case. More precisely, if every Zieve-admissible triple is realizable, then every Zieve-admissible datum with at least four branch points is realizable. In particular, Zieve's conjecture implies the prime-degree conjecture posed by A. L. Edmonds, R. S. Kulkarni and R. E. Stong. Generative-AI disclosure. OpenAI's GPT-5.6 Sol and GPT-6 Astra assisted throughout the research and preparation of this article, including the exploration of reduction strategies, proof development and checking, consistency checks on notation and terminology, and manuscript drafting and revision. The authors reviewed all AI-assisted material with careful scrutiny and take full responsibility for all mathematical claims and for the final text.

We prove that every extremal Kähler–Ricci soliton on a Fano manifold is Kähler–Einstein. This solves the problem of Calamai and Petrecca in full generality.

We prove that Lagrangian mean curvature flow starting from a closed, embedded, cohomogeneity-one Lagrangian in a closed, positive Kähler–Einstein manifold exists for all time, remains embedded, and converges smoothly and graphically to a minimal Lagrangian, under natural exactness and regularity assumptions. We also study the generalised Lagrangian mean curvature flow of Behrndt in Kähler manifolds which are almost-Einstein in the sense that the Ricci form satisfies $\rho = C\omega + ndd^cf$, and which satisfy $C>0$. With analogous assumptions on the flow, we obtain subconvergence to an $f$-minimal Lagrangian submanifold, with an upgrade to smooth graphical convergence in the case that $(M,g,f)$ is analytic. The proof proceeds by first reducing the flow to a weighted curve shortening flow on a compact two-dimensional orbifold. We then establish both a Grayson-type description of finite-time singularities and a long-time subconvergence theorem for weighted curve shortening flow in the orbifold setting. Finally, we use a Łojasiewicz–Simon inequality argument to upgrade to smooth convergence of the full flow.

We prove a rigidity theorem for closed Einstein manifolds of dimension $n\ge6$ under two lower bounds on the curvature operator of the second kind. More precisely, for $L_1, L_2\ge0$ and some real number $\alpha>1$ in a suitable range, we consider \[ \lambda_1\ge-L_1\bar\lambda, \qquad \frac1\alpha\sum_{j=1}^{\alpha}\lambda_j \ge-L_2\bar\lambda. \] Here $\lambda_1\le\cdots\le\lambda_N$ are the eigenvalues of the curvature operator of the second kind $\mathring R$, $N=(n-1)(n+2)/2$, and $\bar\lambda=N^{-1}\sum_{j=1}^N\lambda_j$. Let $\theta(n, \alpha)$ be the constant appearing in \cite[Theorem~1.1]{CW26}; see (2). In the parameter range considered here, $L_2>\theta(n, \alpha)$, so the second condition is strictly weaker than the corresponding condition in \cite{CW26}, while the first condition and that condition do not imply each other. Under an additional explicit relation between $L_1$ and $L_2$, we prove that the manifold is either flat or a spherical space form.

This paper investigates the $L_p$-dual mixed Euclidean-Gauss Minkowski problem associated with the $L_p$-dual mixed Euclidean-Gaussian curvature measure. The solvability of this problem is equivalent to that of a class of Monge-Ampère type equations on the unit sphere. Under smooth assumptions, We establish the existence of solutions for $p<1$, $q<0$, and $q<p$ and uniqueness of solutions for $p\geq q$ to the aforementioned Monge-Ampère type equations, respectively. Furthermore, we obtain a complete solution concerning the existence part of this problem when $p<1$ and $q<0$.

Let $M$ be a complete, non-compact Riemannian manifold. We prove that its Riesz transform is of weak type $(1,1)$, with constant $2$ for real-valued functions. Consequently, it is bounded on $L^p(M)$ for $1<p\leq2$, with constants depending only on $p$, which proves the Coulhon–Duong conjecture. The proof uses an obstacle decomposition for positive self-adjoint operators with sub-Markovian semigroups. Applying this decomposition to the shifted square-root Laplacian and using locality of the Sobolev differential yields the endpoint estimate without geometric or heat kernel assumptions. We also obtain the corresponding result for Dirichlet spaces admitting a local Hilbertian differential calculus.

We construct closed embedded ${\lambda}$-hypersurfaces in ${\mathbb{R}^{n+1}}$ of topological type ${S^1\times M}$ for any ${\lambda \ge 0}$, where ${M\subset\mathbb{S}^n}$ is an isoparametric hypersurface with equal principal curvature multiplicities. This extends previous results of Angenent, McGrath, and Riedler (for ${\lambda = 0}$) as well as Cheng-Wei and Ross (for ${\lambda > 0}$), and in particular completes the missing case of Riedler's construction.

We prove that for all real numbers $\kappa,\tau$ with $\kappa>8\tau^2$ and $\tau\neq 0$, the Berger sphere $\mathbb{E}^3(\kappa,\tau)$ contains smooth closed embedded rotational surfaces of genus one with constant mean curvature which are contained in an open hemisphere. This disproves a conjecture of Fernández and Mira. The main result of this paper is obtained by generative AI, particularly GPT-5.6-sol, Fable 5, and the Danus system, and then verified by the authors.

A Yau-type gradient estimate is proved for positive $p$-harmonic functions on complete Riemannian manifolds with compact boundary, under a Ricci lower bound on the Riemannian manifold and a mean curvature lower bound on the boundary, assuming the Dirichlet condition and a sign condition on the outward normal derivative. The result extends the estimate for harmonic functions by Kunikawa and Sakurai to the full range of $p$-Laplace operators and yields a Liouville theorem when the curvature hypotheses are nonnegative. The cutoff method of the linear theory does not extend when the exponent differs from two, since it requires a directional Hessian of the distance to the boundary that Laplacian comparison cannot control. That cutoff is replaced here by a radial barrier at the boundary and an intrinsic maximum principle in the interior.

For connected closed Lagrangian submanifolds of the homogeneous nearly Kähler sphere $\mathbb S^6(1)$, we first establish a new integral inequality. Then, as applications we prove sharp pinching theorems for the scalar curvature $\tau$ and Ricci curvature Ric, respectively: $\tau \ge \tfrac{23}{8}$ implies either $\tau=6$ with it totally geodesic, or $\tau=\tfrac{23}{8}$ with it congruent to the embedded Berger sphere of Dillen–Verstraelen–Vrancken; whereas ${\rm Ric}\geq\frac18$ implies that the submanifold is either totally geodesic, or has constant sectional curvature $K=\tfrac{1}{16}$, or is congruent to the embedded Berger sphere of Dillen-Verstraelen-Vrancken.

Using N. Honda's construction of twistor spaces, bimeromorphic invarince of $\partial\bar\partial$-lemma for threefolds and embedded resolution, we find a smooth compact complex threefold satisfying the $\partial\bar\partial$-lemma that contains a smooth compact surface for which the lemma fails. This disproves heredity in the smallest possible ambient dimension and yields counterexamples to L. Alessandrini's modification question in every complex dimension at least four. We establish a derived blow-up formula for generalized Bott–Chern complexes with coefficients, compatible with the comparison maps to coefficient cohomology. We also obtain a Dolbeault blow-up formula with bounded derived coefficients. For compact complex manifolds, we introduce a relative $\partial\bar\partial$-property, characterize it using the Frölicher spectral sequence and Hodge filtrations, and prove a blow-up criterion. Finally, counterexamples show the limitations of the classical formulae for coherent coefficients and singular ambient spaces, and the failure of the naive Bott–Chern Künneth formula.

We introduce a geometric structure – a conformal Killing–Yano Ricci soliton (CKY–RS) – that couples conformal Ricci soliton (CRS) geometry to conformal Killing–Yano (CKY) 2–forms. The soliton field of the CRS geometry is given by the divergence of the CKY 2–form. We introduce a conserved CKY–Cotton current and derive a compatibility identity relating the Cotton tensor, the CRS obstruction tensor, and the CKY 2–form. In 4–dimensional Lorentzian signature, we show that, under non-degeneracy and closedness assumptions on the CKY form, a CKY–RS structure forces the conformal representative to be locally Kerr–NUT–(A)dS. For a closed non-degenerate CKY on a Kerr–NUT–(A)dS background, the conformal deformation is necessarily trivial. For Einstein backgrounds of arbitrary dimension and signature, the conformal factor satisfies an eigenvalue equation and an Obata–type Hessian equation. If the background is also compact or a CKY orbit is periodic, the conformal factor is an invariant of the CKY–flow and we obtain simple spectral obstructions to non-trivial CKY–RS structures. From the Hessian equation, we obtain obstruction and classification results for the non-trivial conformal sector, including product/Brinkmann geometries and a Weyl–aligned branch. Finally, we give explicit constructions for static spherically symmetric geometries and BTZ backgrounds, including a CKY–RS realization with a time-dependent conformally flat representative. These results provide a geometric framework for studying CRS with hidden symmetry structure, with potential applications to exact geometries in general relativity.

In this paper we prove in full detail the isomorphism between the solvable group $\mathcal{S}_{2,2+p}$, metric equivalent to the Calabi Vesentini symmetric space $\mathrm{SO(2,2+p)/SO(2)\times SO(2+p)}$, and the solvable group $\mathcal{S}_{\mathrm{L}(-1,p)}$ supporting the Kähler metric of the homogeneous non symmetric special manifold L$(-1,p)$. From an Alekseevskyan point of view the two spaces simply correspond to two different quadratic forms on the same solvable Lie algebra that we present and compare in detail. Furthermore considering the full group of isometries $\mathrm{Iso}_{\mathrm{L}(-1,p)}$ of L$(-1,p)$ we show that $\mathrm{Iso_{L(-1,p)}}\subset \mathrm{SO}(2,2+p)$ is a non-semisimple subgroup of the simple isometry group of Calabi-Vesentini manifolds. Altogether the Special Kähler manifold L$(-1,p)$ can be seen as a coset manifold $\mathrm{Iso_{L(-1,p)}}/\mathrm{H}$ where $\mathrm{H}=\mathrm{U(1)}_L\times \mathrm{SO(p)}$, the generator of $\mathrm{U(1)}_L$ being in $\so(2,2+p)$, yet not in the canonical $\so(2)\oplus\so(2+p)$ subalgebra. The action of $\mathrm{Iso}_{\mathrm{L}(-1,p)}$ on the solvable coordinates of the CV manifold can be constructed directly. This identification provides a valuable tool for Cartan Neural Networks, introducing additional non linear transformations in every map from one layer to the next one of an $r=2$ Neural Network based on the CV Tits Satake universality class; in perspective, this new tool increases expressivity.

Let $N$ be a compact Riemannian manifold whose universal covering supports a convex function. We prove that it admits no quasi-harmonic sphere. No global growth condition is imposed on the convex function.

We study the instability of the naked singularities arising in the spherically symmetric self-similar collapsing of the Einstein–Euler system under gravitational perturbations. We show that small $C^{1,\alpha}$ perturbations (without any symmetries) of the initial conformal metric lead to trapped surface formation. One key point is that the speed of sound is strictly slower the the speed of light, so the gravitational radiation can become sufficiently concentrated to form a trapped surface before the fluid develops any singularity.

Riemannian groupoids describe Riemannian foliations together with their symmetries. In this article, we extend classical Molino's theory, which concerns the structure of Riemannian foliations on compact manifolds, to the setting of regular Riemannian groupoids with compact connected object manifolds. We observe that the orbit foliation associated with a regular Riemannian groupoid defines a Riemannian foliation on the object manifold. The main result shows that the normal representation of a Riemannian groupoid extends naturally to an action on Molino's structures of the orbit foliation, thereby yielding the fundamental structural description of a regular Riemannian groupoid. In addition to the main result, we clarify the essential role played by basic Lie algebroids in Molino's theory and identify the normal representation with the degree-one cohomology representation of the adjoint representation up to homotopy. We also establish comparison results under Riemannian Morita equivalence. For regular Riemannian groupoids, we give sufficient conditions for the associated basic Lie algebroids to become isomorphic after pullback to a common refinement.

We construct a sequence of stationary weak Yang-Mills connections on the trivial vector bundle $B^5\times\mathbb{R}^{15}$ with structure group $\operatorname{SO}(15)$ whose weak limit is a nonstationary weak Yang-Mills connection. Motivated by the stationarity question arising in Tian's work on Yang-Mills compactness, our result establishes the failure of weak closure for stationary weak Yang-Mills connections with an isolated singularity. It also provides a Yang-Mills analogue of the phenomenon exhibited by Ding, Li and Li for harmonic maps.

It is conjectured that a complete, orientable Einstein four-manifold with strictly positive sectional curvature must be isometric to either the round four-sphere $S^4$ or the complex projective plane $\mathbb{CP}^2$ with the Fubini–Study metric. In this paper, we confirm this conjecture when the manifold is homeomorphic to $S^4$ or $\mathbb{CP}^2$. More precisely, we show that a complete, orientable Einstein four-manifold $M$ with strictly positive sectional curvature and Euler characteristic $\chi(M) \le 3$ (equivalently, by Freedman's classification theorem, homeomorphic to $S^4$ or $\mathbb{CP}^2$) is isometric to either the round $S^4$ or the Fubini–Study $\mathbb{CP}^2$. As an application, we obtain that a complete, orientable Einstein four-manifold with strictly positive sectional curvature admitting a nontrivial Killing field is isometric to either the round $S^4$ or the Fubini–Study $\mathbb{CP}^2$.

We prove congruence theorems for capillary hypersurfaces of Euclidean space supported on a hyperplane or on a sphere, extending a result of Ros for closed hypersurfaces. We also establish a Heintze–Karcher type inequality for capillary hypersurfaces in substatic manifolds.

We prove a boundary analogue of the rigidity theorem for Ricci-pinched three-manifolds under a superquadratic volume-growth assumption. More precisely, we show that a complete connected three-manifold with convex boundary, pinched Ricci tensor, and pinched second fundamental form is isometric to the Euclidean half-space. The proof is potential-theoretic. We construct and study mixed Dirichlet–Neumann $p$-capacitary potentials, establish monotonicity formulas for their free-boundary level sets, and use a weak Gauss–Bonnet formula adapted to this setting.

We generalize de Rham's decomposition theorem to proper metric spaces: every proper metric space has a unique decomposition as the product of a Euclidean space and a finite or countable pointed $\ell^2$-product of nontrivial irreducible non-Euclidean factors. More generally, every complete metric space splits canonically into a Hilbert factor, a pointed $\ell^2$-product of irreducible non-Euclidean factors, and a factor with no nontrivial irreducible factors. These results partially answer a question of Ulam.

For a smooth, connected, oriented surface $\Sigma$, topologically a sphere, immersed in the homogeneous space $E(\kappa,\tau)$ with $\tau\ne0$, we show that a one-sided Cauchy–Riemann-type inequality on the original Abresch–Rosenberg differential $\mathcal{Q}_{AR}$ – much weaker than requiring $\mathcal{Q}_{AR}$ to be holomorphic – already forces the mean curvature $H$ to be constant. The proof is uniform across every $E(\kappa,\tau)$, including the round sphere, and combines the Bers–Vekua similarity principle with the Poincaré-Hopf index formula for line fields; it is logically independent of the algebraic argument used, in the companion paper~\cite{AlencarRosenberg2026}, to characterize CMC immersions by holomorphy of $\mathcal{Q}_{AR}$ alone. We also show that the topological hypothesis is sharp: on every Berger sphere, including the round one, there exist compact non-CMC tori – the preimages under the Hopf-type submersion $\pi:E(\kappa,\tau)\to M^2(\kappa)$ of simple closed curves in the base with nonconstant geodesic curvature – satisfying the same inequality with a constant bound. As an elementary consequence of the classical Poincaré-Hopf theorem for line fields, we record that every closed surface on which $\det S<0$ everywhere – a condition satisfied, in particular, by every Hopf tube, where $\det S\equiv-\tau^2$ – is a torus.

The critical dimension for free maps on m-manifolds is $q_m=m(m+3)/2$. We show that $Free^\infty(\mathbb{T}^m,\mathbb{R}^{q_m})\neq\emptyset$ for every $m\ge 1$. The main tool is a product construction: given free maps in critical dimension on $\mathbb{T}^a$ and on $\mathbb{T}^b$, together with a cross-free map on $\mathbb{T}^a\times \mathbb{T}^b$, whose mixed Hessian is invertible everywhere, we obtain a free map in critical dimension on $\mathbb{T}^{a+b}$, the osculating matrix of the product being block-triangular. Cross-free maps are additive in each argument, and in the torus setting none exists when one factor is one-dimensional. We construct explicit cross-free maps on $\mathbb{T}^2\times \mathbb{T}^2$ and on $\mathbb{T}^3\times\mathbb{T}^4$ using quaternionic multiplication. Combined with the low-dimensional cases $m\leq 5$, constructed explicitly by the author in a previous publication, these yield the result by induction.

Graph neural networks are widely used for drug–target affinity (DTA) prediction, and discrete Ricci curvature has recently been used to characterize molecular graph geometry. Existing curvature-aware DTA approaches mainly use static curvature on the drug graph while representing proteins primarily with sequence-derived features. This leaves pair-adaptive use of graph geometry underexplored, which may limit adaptation to unseen entities in cold-start settings relevant to practical screening. We present CurvFlow-DTA, which replaces a single static curvature representation with weighted Forman curvature flow on both molecular and protein residue–residue contact graphs. A label-independent flow trajectory is precomputed for each entity, and a pair-conditioned selector determines the horizons read by a dual-branch Flow-GINE. A frozen ESM-2 supplies residue-level representations and contact scores used to construct the protein graph. Inference requires only SMILES strings and protein sequences, without a bound complex structure. On Davis and KIBA, CurvFlow-DTA improves on the protocol-matched Ricci-GraphDTA baseline in every warm and cold-start setting. Warm-split mean squared error (MSE) decreases by $19.9\%$ on Davis and $18.9\%$ on KIBA. Across the six cold-start comparisons, MSE decreases by $14.3$–$27.4\%$, with higher concordance index (CI) throughout. Within our compiled set of literature baselines, CurvFlow-DTA achieves the lowest MSE on both warm benchmarks and across four out of six cold-start evaluation settings.

We classify compact Kähler–Einstein surfaces whose metric is induced by a holomorphic isometric immersion into a finite-dimensional complex projective space. No symmetry assumption and no bound on the codimension are imposed. We prove that the only such surfaces are \[ (\PP^2,m g_{\FS}) \quad\text{and}\quad \bigl(\PP^1\times\PP^1,m(g_{\FS}\oplus g_{\FS})\bigr), \qquad m\in\mathbb Z_{>0}, \] realized respectively by the Veronese and Segre–Veronese embeddings. The main new ingredient is a codimension-independent exclusion of the entire Fano-index-one branch, combining a common anticanonical root construction with Gram–Gauss rank estimates and, in degree five, an equivariant curvature argument. Consequently, every connected compact Kähler–Einstein surface whose metric is induced by a holomorphic isometric immersion into a finite-dimensional complex projective space is homogeneous.

We prove a Pólya–Szegö inequality for functions defined on an $n$-dimensional submanifold $\Sigma$ of a complete noncompact Riemannian manifold with nonnegative sectional curvature. The associated rearrangement is a Schwarz rearrangement on $\mathbb R^n$, and the constant depends on the $L^n$-norm of the mean curvature of $\Sigma$ and an isoperimetric quantity obtained by Brendle. As applications, we derive Sobolev, Log-Sobolev, Hardy, and Gagliardo–Nirenberg inequalities on submanifolds of arbitrary codimension under a small total mean curvature assumption. In the critical Sobolev case, we obtain Moser–Trudinger inequalities on finite-volume submanifolds and exact growth inequalities on submanifolds with infinite volume. Under suitable assumptions, the Pólya–Szegö constant equals one; in this case, the critical constants in the inequalities coincide with the sharp Euclidean ones.

We study the log-concavity property of first Dirichlet eigenfunctions on domains in $\mathbb{CP}^2$. For every smooth $1$-convex domain $\Omega\subset\mathbb{CP}^2$, we prove the quantitative estimate \[ \nabla^2(-\log u) > \max\left\{\psi(s),\frac85\right\}g, \text{ where } \psi(|\grad f|^2) = \frac{s}{\sqrt{1+s}}-\log(1+s), \] for its first Dirichlet eigenfunction $u$. In particular, $u$ is strictly log-concave. As consequences, we obtain a uniform convexity estimate for the regular level sets of $u$ and the fundamental gap bound $\lambda_2-\lambda_1>46/5$.