Papers

Given independent and identically distributed samples samples from a probability distribution in a potentially high-dimensional real space, we study the problem of testing whether the distribution is concentrated near a real algebraic complete intersection of prescribed dimension, bounded degree, and bounded condition number. We design an explicit and effective learning procedure which either certifies the nonexistence of such a manifold, up to a controlled relaxation of the approximation threshold, or returns a candidate regression manifold with controlled geometric complexity. Equivalently, the procedure tests the manifold hypothesis within this hypothesis class. The proposed procedure relies on quantitative geometric estimates for regular polynomial systems, which lead to a tractable auxiliary optimization problem. We then develop a data-driven algorithm to solve this auxiliary optimization problem, establishing explicit bounds on its sample and arithmetic complexity.

We discuss the recently introduced concept of hyperquivers, due to Muller, Nanda, and Seigal, from the point of view of Kähler and hyperkähler quotients. In particular, we show that the Kähler hyperquiver variety ${\mathcal M}$, corresponding to a hyperquiver with a single hyperedge, is a projective variety isomorphic to the geometric quotient of the space of tensors with maximal multilinear rank. Moreover, the hyperkähler hyperquiver variety corresponding to doubling of this hyperquiver can be identified with the linear scheme corresponding to the sheaf of reflexive $1$-forms on ${\mathcal M}$.

We construct smooth closed embedded minimal surfaces with everywhere negative Gaussian curvature in the unit round four-sphere. More precisely, for every sufficiently large integer $m$, our construction gives a minimal surface $\Sigma_m$ of genus $m^2+1$ such that $\operatorname{Area}(\Sigma_m)\to 4\pi^2$ as $m\to+\infty$. This result yields a complete solution to Problem 101 in Yau's 1982 list of open problems.

This note gives direct proofs that, under an effective isometric $T^2$ action, Einstein metrics on smooth homotopy four-spheres are round and complete Ricci-flat metrics on smooth manifolds homeomorphic to $\mathbb R^4$ are Euclidean or positive one-center Taub–NUT, up to isometry, scaling, and orientation reversal.

We prove that every weakly pseudoconvex Oka manifold admitting a positive line bundle is elliptic in the sense of Gromov, thereby proving Gromov's ellipticity conjecture. The proof establishes a cohomological construction of local sprays. Given a bundle morphism $\varphi\colon E\to T_X$, we introduce quadratic $\varphi$-vector fields on $E$ whose flows yield local sprays with fibre derivative $\varphi$. Their existence is characterized by the vanishing of the symmetrized $\varphi$-Atiyah class of $E$. This gives a local dominating spray on every weakly pseudoconvex manifold admitting a positive line bundle, without any Oka assumption. The positivity hypothesis is essential even for local existence: blow-ups at a single point of complex tori of algebraic dimension zero, and Kummer surfaces of algebraic dimension zero, admit no local dominating spray. The torus examples give compact Kähler Oka manifolds that are not elliptic and show that ellipticity is not preserved under blowing up, and that it is neither open nor closed in holomorphic families of compact Oka manifolds. Combined with the work of Xie and Zhao, the Kummer examples yield Oka K3 surfaces that are not elliptic. Under the additional Oka assumption, we globalize the local dominating sprays constructed above. The proof combines Oka approximation with a gluing argument based on weighted $L^2$ estimates for the $\bar\partial$-equation.

Let $M$ be a smooth manifold and $TM$ its tangent bundle. A $C^0$-Finsler structure of $M$ is a continuous function $F: TM \to [0, \infty)$ such that restricted to each tangent space is an asymmetric norm. A mollifier smoothing of $F$ is a family of Finsler structures $F_\varepsilon: TM \to [0, \infty)$, parameterized by $\varepsilon > 0$ and constructed using the standard mollifier, such that $F_\varepsilon \to F$ uniformly on compact subsets as $\varepsilon \to 0$. In this work, we construct two mollifier smoothings of $F$. In the first, we assume that $F: TM \to [0, \infty)$ is strongly convex, that is, that $F$ restricted to each tangent space is strongly convex. In the second, we assume that $M=G$ is a Lie group and that $F:TG \to [0, \infty)$ is strongly convex and left-invariant, and in this case, $F_\varepsilon$ is also left-invariant. In both cases, we prove that if $F$ is a Finsler structure, objects such as the fundamental tensor, the Chern, Cartan, Hashiguchi and Berwald connections, and the flag curvature of $(M, F_\varepsilon)$ converge uniformly on compact subsets to the corresponding objects of $(M, F)$ as $\varepsilon \to 0$.

It was known that the canonical bundle of a smooth toroidal compactification of a noncompact finite-volume ball quotient is nef in dimension $n\geq3$ and ample in dimension $n\geq6$. We prove that the canonical bundle is ample in dimension $n\geq4$ and show that this dimension bound is sharp. We construct infinitely many pairwise noncommensurable noncompact Picard modular threefolds whose smooth projective toroidal compactifications have non-ample canonical bundle. Each compactification contains a smooth rational curve meeting the interior and having canonical degree zero. More generally, in every dimension $n\geq2$, we construct infinitely many pairwise noncommensurable noncompact Picard modular varieties whose smooth toroidal compactifications contain smooth rational curves meeting the interior. In particular, such curves persist in arbitrarily high dimension, even when the canonical bundle is ample.

The oscillator groups are four-dimensional solvable Lie groups, extensions of the Heisenberg Lie group. We present them as circle bundles of the standard contact sub-Riemannian structure on $\mathbb{R}^{3}$. We define sub-Riemannian and sub-Lorentzian structures on them, describe their geodesics explicitly and determine which of them are periodic. We also study a generalization: Let $G$ be a semisimple Lie group and $K$ a compact subgroup such that $\left( G,K\right) $ is a symmetric pair of the compact or the noncompact type. Let $\mathfrak{g}$ and $% \mathfrak{k}$ be the Lie algebras of $G$ and $K$, respectively. We consider on $\mathfrak{g}$ the usual nilpotent Lie group structure with center $\mathfrak{k}$ and call it $N\left( \mathfrak{g},\mathfrak{k}\right)$. A suitable semidirect product with $\operatorname{Ad}\left( K\right) $ yields a solvable Lie group $\operatorname{Osc}\left( G,K\right) $, which generalizes the oscillator groups and is also quadratic (that is, it possesses bi-invariant metrics; in particular, a canonical one). We define nonholonomic pseudo-Riemannian structures on it and find their geodesics explicitly. In doing so, we obtain a result that may not be merely auxiliary: formulas for the monoparametric subgroups of $\operatorname{Osc}\left( G,K\right) $ and for the sub-Riemannian geodesics of $N\left( \mathfrak{g},\mathfrak{k}\right) $ with the standard left-invariant distribution $\mathcal{D}$. Moreover, when $G/K$ is compact with rank one, we realize $\operatorname{Osc}\left( G,K\right) $ as a Stiefel bundle of $\left( N\left( \mathfrak{g},\mathfrak{k}\right) ,\mathcal{D}\right) $, up to finite coverings.

Let $(M^{2m},g,f,J)$ be a complete nonsteady gradient Kähler-Ricci soliton satisfying $\mathrm{Ric}+\nabla^2 f=\lambda g$, $\lambda\neq 0$. We prove that constant scalar curvature forces the soliton to be rigid. More precisely, the universal cover splits holomorphically and isometrically as $N^{2k}\times\mathbb{C}^{m-k}$, where $N^{2k}$ is Kähler-Einstein with $\mathrm{Ric}_{g_N}=\lambda g_N$. In the shrinking case the quotient is trivial; in the normalization $\lambda=1/2$ one has $R\equiv k$. The Kähler result rests on a Riemannian rigidity criterion. On a complete nonsteady gradient Ricci soliton, if $\mathcal{L}_{\nabla f}\mathrm{Ric}$ is nonnegative or nonpositive everywhere, then it vanishes and the soliton is rigid; no assumption on the scalar curvature is needed. Along the Ricci flow generated by the soliton, this means that a Ricci tensor that is monotone in time is constant in time, and that this forces rigidity. We also give a direct proof that the pinching $0\leq\mathrm{Ric}\leq\lambda g$ forces constant scalar curvature and radial flatness, yielding the rigidity conclusion through the Petersen-Wylie characterization. The proof combines a weighted cutoff argument with a partial Codazzi symmetry for the Ricci endomorphism.

We prove the Hawking and Penrose singularity theorems for Lorentz–Finsler structures of mixed horizontal/vertical regularity $C^{(1,2)}$ in the $C^1$ weighted case and $C^{(1,3)}$ in the unweighted case. At this regularity, geodesics need not be uniquely determined by their initial conditions, and the (weighted) Ricci curvature must be interpreted distributionally. Our approach relies on a two-stage approximation procedure: first, a homogeneity-preserving convolution, which is of independent interest in the broader semi-Riemann–Finsler setting, and second, a causality-adapted approximation à la Chruściel–Grant. In the special case of $C^1$ Lorentzian metrics, our results extend the $C^1$ singularity theorems of Graf to the $C^1$ weighted case.

In this paper we study the geometrical entities as Gauss and mean curvatures attached to the ruled normal surfaces associated with rectifying curves. A particular and special example of rectifying curves is also investigated, via its attached ruled normal surface. Our main result is the proof of the non-existence of rectifying curves that are simultaneously Bertrand curves.

For a complex simple Lie algebra $\mathfrak{g}$ with compact conjugation $\tau$, Killing form $B$, and Hermitian form $H(x, y) = -B(x, \tau y)$, it is shown that $H([x,y],[x,y]) \leq (h^{\vee})^{-1}H(x,x)H(y,y)$, where $h^{\vee}$ is the dual Coxeter number, with equality exactly for $H$-orthogonal pairs in $\tau$-stable long-root subalgebras isomorphic to $\mathfrak{sl}(2, \mathbb{C})$. Equivalently, the comass of the Cartan $3$-form with respect to $H$ equals its comass on the compact real form: the spectral norm of the Cartan $3$-form does not increase under complexification. For the Frobenius norm on the image of a representation of Dynkin index $\ell$ the optimal constant is $2/\ell$; this contains the Böttcher–Wenzel and Bloch–Iserles inequalities, extends the latter to complex skew-symmetric matrices, and improves them substantially for the exceptional Lie algebras of types $F_{4}$, $E_{6}$, $E_{7}$, and $E_{8}$. The proof uses the Nahm algebra of $\mathfrak{g}$ with the conjugation induced by $\tau$: critical points of the relevant function correspond to $\tau$-twisted idempotents, and its Hessian is governed by $\tau$-twisted multiplication operators, whose anticommutation with multiplication by $i$ turns the one-sided second order condition at a maximum into a two-sided spectral bound. Subalgebras isomorphic to $\mathfrak{sl}(2, \mathbb{C})$ of Dynkin index $j$ give critical points with critical value $1/(jh^{\vee})$, whose Morse indices are computed, and the set of maximum points is a nondegenerate critical manifold. Geometrically, the maximal complex sectional curvature of a compact simple Lie group equals its maximal sectional curvature.

In this note, we show that a locally conformally flat free boundary minimal hypersurface in $B^n$ must be either an equatorial ball or a critical catenoid for all $n \geq 4$. As a consequence, we conclude that the equatorial ball is unique among locally conformally flat free boundary minimal hypersurfaces in $B^n$ with the topology of $B^{n - 1}$.

In this paper, we study compact vacuum static spaces of dimension $\ge 4$ with nonnegative sectional curvature. First, we prove that an $n$-dimensional compact vacuum static space is isometric to a standard sphere provided it has positive sectional curvature and the complete divergence of its Weyl curvature tensor is nonnegative. Under the slightly weaker condition of nonnegative sectional curvature, we show that such a vacuum static space must have parallel Ricci curvature tensor. Second, we examine the critical point equation arising from the critical metrics of the total scalar curvature functional on the space of Riemannian metrics restricted to unit volume and constant scalar curvature. We demonstrate that a compact Riemannian manifold with nonnegative sectional curvature and nonnegative complete divergence of the Weyl curvature tensor, which admits a nontrivial solution to the critical point equation, must be Einstein and is isometric to a standard sphere. Our results can be viewed as extensions of the results in [4].

We study Killing vector fields, Ricci solitons and hyperbolic Ricci solitons on the hyperbolic strip $S_a=\{(x,y)\in\mathbb{R}^2:0<y<a\}$ endowed with the metric $g_{S_{a}}=\frac{\pi^2}{a^2\sin^2(\pi y/a)}(dx^2+dy^2)$, and on its unit tangent bundle $T^1S_a$ endowed with the Sasaki metric $g^S$. On the strip, we show that the Gauss curvature is constant equal to $-1$, that the Lie algebra of Killing fields is isomorphic to $\mathfrak{sl}(2,\mathbb{R})$, that every Ricci soliton is trivial (Killing potential, $\lambda=-1$), and that a $2$-Killing field which is conformal, or which is the potential of a hyperbolic Ricci soliton with $\lambda\neq0$, is a Killing field. On the unit tangent bundle, we prove that the Lie algebra of Killing fields of $(T^1S_a,g^S)$ is four-dimensional and isomorphic to $\mathfrak{sl}(2,\mathbb{R})\oplus\mathbb{R}$, that $g^S$ is not Einstein but is $\eta$-Einstein with constant coefficients, $\operatorname{Ric}^S=-\tfrac32\,g^S+2\,\eta\otimes\eta$, and that $(T^1S_a,g^S)$ admits no Ricci soliton. For hyperbolic Ricci solitons we prove non-existence for Killing, left-invariant, and $\theta$-invariant potentials with conformal projection, and we reduce the general $\theta$-invariant case to a system of equations on the strip; the classification for an arbitrary potential is left open. Finally, we show that the $\eta$-Ricci solitons of $g^S$ have a Killing potential and $\lambda=\tfrac32$, $\mu=-2$, and that the $\eta$-hyperbolic Ricci solitons obtained in the same classes of potentials are the trivial ones, with a Killing potential, $\mu=-\tfrac32$ and $\nu=2$.

We investigate weighted perimeters of convex bodies in $\mathbb{R}^n$. Our main result states that the weighted perimeter is continuous with respect to the Hausdorff metric for every radial weight function. We also establish monotonicity under inclusion under suitable assumptions on the radial weight, and show that these assumptions are sharp. These results are applied to show the existence of minimizers for a shape optimization problem.

Let $M^n\subset S^{n+m}$ be a compact minimal submanifold of the unit sphere, and let $\lambda_1\geq\cdots\geq\lambda_m\geq0$ be the eigenvalues of its fundamental matrix. We prove that $\sum\limits_{\alpha=1}^{\min\{n,m\}}\lambda_\alpha+\lambda_2\leq n$ forces $M$ to be a totally geodesic subsphere, a generalized Clifford torus, or a Veronese manifold. We first show that equality everywhere in the pinching condition forces the second fundamental form to be parallel. Next, by computing the eigenvalues of the fundamental matrices of irreducible symmetric $R$-spaces, we obtain a complete classification of the equality cases among minimal submanifolds with parallel second fundamental form.

We establish local volume doubling and a local $L^2$-Poincaré inequality for metric measure spaces tamed by a measure in the extended Kato class that satisfy the bounded interpolation property. To this end, we construct a bi-Lipschitz time change to a metric measure space satisfying a uniform lower Ricci curvature bound. As an application, we obtain pointed measured Gromov-Hausdorff precompactness for such spaces.

In this paper, we study the global regularity of oblique boundary value problems for n-1 type augmented Hessian equations on a bounded domain, without imposing any convexity condition either on the domain or on the matrix-valued function in the n-1 type augmented Hessian equations. We establish a global existence and uniqueness theory for classical elliptic solutions by deriving global a priori estimates up to second-order derivatives. Besides the known applications for n-1 type Monge-Ampere operators in optimal transportation and geometric optics, the general theory here embraces equations raised by Li-Sheng, Some Dirichlet problems arising from conformal geometry. Pac. J. Math. 251, 2011, 337-359 and Guan-Qiu-Yuan, Fully nonlinear elliptic equations for conformal deformations of Chern-Ricci forms. Adv. Math. 343, 2019, 538-566.

This paper addresses the design of flatness-based tracking controllers for backward-flat nonlinear discrete-time systems. First, we show that backward-flat discrete-time systems can be constructively obtained from differentially flat continuous-time systems by applying an implicit Euler discretization to a structurally flat triangular representation. Subsequently, two approaches for the exact linearization of backward-flat systems are considered. Besides a dynamic feedback based on prelongations, we show that a linearizing feedback involving lower-order backward-shifts of the flat output can be employed without explicitly implementing the corresponding dynamic extension. This allows the order of the resulting tracking error dynamics to be reduced while requiring only stored values of past system trajectories. Based on the resulting linear input-output representation, flatness-based tracking control laws are derived. The proposed discretization and controller design are illustrated for a 2D gantry crane.

We prove a sharp Hessian inequality for the natural comparison Green's function of an Einstein manifold with positive sectional curvature, which may be viewed as a Green's function analogue of the Hessian comparison theorem and as a positively-curved, elliptic counterpart to the matrix Li-Yau-Hamilton inequality for the heat equation. We also show that this inequality is closely related to a family of monotonicity formulae, extending previous results of Park to the setting of positive curvature. We further present several geometric applications, such as a sharp comparison inequality for the sum of two distinct Green's distance functions.

We study homogeneous sub-Riemannian manifolds whose normal extremals are orbits of one-parameter subgroups of the acting group. Our first main result shows that every sub-Riemannian geodesic orbit manifold satisfies the Goh condition and consequently admits no strictly abnormal length minimizers. Our second main result classifies all compact, connected, simply connected sub-Riemannian geodesic orbit manifolds with nonabelian simple isotropy group, thereby extending the Riemannian classification of Chen, Nikolayevsky and Nikonorov.

We study gradient Ricci solitons whose potential functions have level hypersurfaces with parallel second fundamental form. We show that such a soliton is locally a multiply warped product of a one-dimensional base and Einstein fibers, with potential function depending only on the base. Thus, locally, this condition characterizes the multiply warped structure that occurs in many classical constructions of Ricci solitons. We further prove that the same local structure follows for any gradient Ricci soliton if just one regular level hypersurface has parallel second fundamental form and parallel intrinsic Ricci tensor. As an application, we prove that a complete nonsteady gradient Ricci soliton with constant scalar curvature is rigid as soon as one regular level hypersurface has this property. In particular, the conjecture of Cao holds for gradient shrinking Ricci solitons with this property. We also prove that a complete gradient shrinking Ricci soliton whose regular level hypersurfaces have parallel second fundamental form and at most one nonzero principal curvature is rigid, without any assumption on the scalar curvature.

We give a convergent explicit representation of the Ricci-flat Kähler metrics supplied by the Calabi–Yau theorem in a one-parameter family of classes on a fixed Kummer K3 surface. An explicit radial background has normalized Monge–Ampère residual \(O(a^4)\). Its scalar Green operator \(G_a\) is represented by periodic Ewald kernels, separated radial kernels, finite-dimensional Schur complements, and convergent Neumann series. The coefficients are defined by \(U_{a,1}=-G_af_a\) and \(U_{a,n}=G_a\sum_{j=1}^{n-1}Q_a(U_{a,j},U_{a,n-j})\), where \(Q_a\) is the polarized quadratic Monge–Ampère term. A Green estimate of order \(a^{-1}\) gives a first correction of order \(a^3\) and a convergence parameter of order \(a^2\). A Catalan majorant proves absolute convergence on the entire smooth surface for every sufficiently small fixed \(a\). The sum defines a positive form solving the volume equation; comparison identifies the sum with the smooth normalized Calabi–Yau potential. We also record sufficient analytic conditions for the same recursion on collapsing elliptic K3 surfaces.

In this paper, we first classify complete four-dimensional shrinkers under a pinching condition on the $2$nd-Ricci curvature. We then classify closed four-dimensional shrinkers with (weighted) integral parameter-pinching of the half Weyl curvature under several pinching assumptions on the biorthogonal curvature. Finally, we establish a Hitchin-Thorpe type inequality for closed non-trivial four-dimensional shrinkers under some assumptions on the half Weyl curvature and the biorthogonal curvature.

Let $(M^n,g)$, $n\ge2$, be a compact connected Riemannian manifold with smooth nonempty boundary, $\operatorname{Ric}_M\ge(n-1)Kg$ for $K>0$, and nonnegative outward boundary mean curvature. We derive an explicit algebraic lower bound $B_\eta$ for the first Dirichlet eigenvalue from Kasue's mixed inradius model. With $R$ the inradius and $s=4KR^2/\pi^2$, the bound involves one square root and approximates the model eigenvalue with relative error $O_n(s^4)$ as $s\downarrow0$ for fixed $n$. The construction uses the first two moments of a positive representation of the averaged model potential and a sharp pointwise corrector estimate. For the chosen spectral transfer, the first two moments determine an optimal bound. We also show that the model eigenvalue is the geometric infimum at fixed $(n,K,R)$. For $0<s<1$, it is approached by smooth collapsing balls with sectional curvature at least $K$ and totally geodesic boundary. At the maximal inradius, the model value is attained by the round hemisphere. The bound strictly improves the variance refinement throughout $0<s\le1$. We also give a direct algebraic comparison with an optimized power-test benchmark.

We prove that the geodesic flow of a closed real-analytic Riemannian manifold has zero topological entropy if it admits globally defined, real-analytic Liouville integrals. This proves a longstanding conjecture proposed by Bolsinov and Taimanov.

In this paper we investigate adapted complex polarizations associated with left-invariant Koszul connections on real Lie groups. Adapted complex polarizations arise as a natural generalization of a complex structure on a manifold. We investigate the existence of such polarizations on the tangent bundle of certain Lie groups with left-invariant connections. We use a practical criterion for the existence of global adapted complex polarizations in terms of the $i\mathbb R$-completeness of the geodesic flow of the connection. In the Lie case, the Euler-Arnold vector field determines the behaviour of the geodesic flow. By combining what we know about the Euler-Arnold vector field of certain Lie groups and the $i\mathbb R$-completeness criterion, we provide several examples of left-invariant ac-polarizations defined on the tangent bundle of Lie groups. Along with pseudo-Riemannian metric connections, we discuss biinvariant connections in several cases. Our focus is on solvable Lie groups, because non-solvable Lie groups are better understood. The main result of this paper is that we demonstrate several not locally symmetric but $\mathcal P$-complete connections.

We study the relation between the saddle property and local metric minimality for smooth surfaces in Euclidean spaces. Anton Petrunin and Stephan Stadler proved that every smooth strictly saddle surface in $\mathbb{R}^3$ is locally metric-minimising and expressed the expectation that this phenomenon fails in $\mathbb{R}^4$. We confirm this expectation and prove a stronger density result: Every smooth strictly saddle embedding of a closed disc into $\mathbb{R}^4$ can be approximated arbitrarily closely in the $C^\infty$-topology by smooth strictly saddle embeddings that are nowhere locally metric-minimising. In particular, nowhere locally metric-minimising saddle embeddings exist in $\mathbb{R}^d$ for every $d \geq 4$. The main ingredient is the construction of non-trivial infinitesimal isometric deformations of saddle immersions in $\mathbb{R}^4$ supported in arbitrarily small discs. We also provide an explicit twisted quadratic surface that is saddle in a neighbourhood of the origin but fails to be locally metric-minimising there, by deriving a necessary condition for metric minimality.

We investigate the symplectic geometry of homogeneous spaces associated with semisimple Lie groups, with particular emphasis on cotangent bundles of generalized flag manifolds. Our approach gives an explicit description of the Liouville symplectic form on these spaces in terms of the connections and curvature forms of principal bundles naturally arising from the Lie group structure. This provides a geometric link between the Lie-theoretic properties of generalized flag manifolds and the symplectic structures carried by their cotangent bundles.