Papers

In this paper, we introduce some type vector fields with respect to a semi-symmetric non-metric (SSNM) connection. We investigate several geometric properties of a K-contact manifold equipped with an SSNM connection and provide a concrete example to justify the relation between the scalar curvature of the SSNM connection and Levi-Civita connection that we have obtained in this paper. Furthermore, we have found the nature of Riemann solitons, conformal Ricci solitons and conformal $\eta$-Ricci-Yamabe solitons on K-contact manifolds admitting a SSNM connection.\\ Finally, we determine the necessary and sufficient conditions for such a manifold to be $\Tilde{\tau}$-semi-symmetric, quasi-conformal-semi-symmetric and pseudo-projective-semi-symmetric.

For the $k$-self-similar naked singularity solutions of Einstein–scalar field equations constructed by Christodoulou, we construct a family of interior gravitational perturbations leading to trapped surface formation that remain large but finite at the threshold Hölder regularity, while converging to zero in all regularities below the threshold. This extends the interior spherically symmetric result below the threshold in \cite{Li25} to non-spherically symmetric setting. We further construct a genuinely localized family of perturbations whose angular support shrinks to a single point, with smooth convergence to zero away from that point. This exploits the additional angular freedom available beyond spherical symmetry. Moreover, when measured instead in Sobolev regularity, such genuinely localized family of perturbations still converges to zero in all regularities below the same threshold.

We study the iterated action of the Lie derivative on the curvature tensor of a Riemannian manifold with rank-one anisotropic curvature, whose Riemann tensor is expressed via the Kulkarni-Nomizu product as $R = \lambda (\xi^\flat\otimes\xi^\flat)\owedge g$. First, we examine the conditions under which such a manifold admits a Ricci soliton structure and demonstrate that this property implies the almost $\theta$-Yamabe soliton structure. Furthermore, we show that if the associated potential vector field $X$ is a symmetry of the Ricci tensor of a fixed order $k$ (i.e., $\mathcal{L}_X^k \operatorname{Ric} = 0$), the geometric problem reduces to solving a partial differential equation of order $k+1$ along the flow. Finally, under the assumption that $X$ is a conformal vector field ($\mathcal{L}_X g = 2\varphi g$) whose infinitesimal flow preserves the line distribution $\mathcal{D}=\operatorname{Span}\{\xi\}$ (with $[X,\xi]=a\xi$ for $a\in\mathbb{R}$), we prove that several key geometric problems (such as establishing the relation $\mathcal{L}_X^k R = R$, determining the minimal order $k$ for $X$ to be a Lie curvature symmetry, or satisfying $\mathcal{L}_X^{k+1}R = f \mathcal{L}_X^k R$ for a continuous function $f$) are equivalent to a scalar differential problem governed by the operator $D_X = X + 6\varphi + 2a$.

Let $X$ be a compact Kähler manifold and $(\alpha,\beta)$ be a pair of Kähler classes. We show that when $(\alpha,\beta)$ is $J$-semistable, the collection of positive-dimensional irreducible $J$-null subvarieties is finite. We also obtain a uniform positive lower bound for the normalized $J$-slope of every irreducible non-$J$-null subvariety. This confirms the finiteness and uniform-gap expectation of Khalid-Sjöström Dyrefelt \cite[Section 1.2, Theorem 1.5; Remark 1.8]{KSD26}. As a corollary, we show that the collection of optimal destabilizers is finite when the stability threshold is nonpositive.

Motivated by the role of the Wigner caustic in semiclassical phase-space analysis, we extend it, together with the centre symmetry set, from regular planar curves to cooriented frontals. For an angularly regular parallel pair, the signed speed of the Wigner caustic is one half of the difference of the extended signed radii of curvature, while singular points of the finite centre symmetry set are the critical points of their projective ratio. These formulas yield criteria and explicit invariants for ordinary and higher cusps, an order-lowering relation between the two constructions, and a multiplicity-weighted extension of the classical cusp-count inequality for strictly convex ovals. We also analyse parallel pairs containing a singular frontal which is not a front. A $5/2$-cusp produces either a $5/2$- or, at a signed-radius resonance, a $5/3$-cusp on the Wigner caustic. The opposite resonance sends the centre symmetry set to infinity. We obtain the corresponding transfer results for a $5/3$-cusp and give a projective completion which resolves vanishing denominators and simultaneous finite-order zeros of the two signed radii.

We prove a general boundary gluing principle for families of Riemannian metrics with pointwise curvature restrictions on manifolds with possibly non-compact boundary. This includes, as special cases, the classical gluing theorems by Perelman for metrics with positive Ricci curvature and convex singularities and by Gromov-Lawson and Bär-Hanke for metrics with positive scalar curvature and mean convex singularities. Using the language of algebraic curvature cones, our theorem implies and unifies many more previous gluing results. We also prove a boundary deformation principle that enables us to compare spaces of metrics with interior curvature restrictions and different boundary conditions. Our construction is based on the local flexibility lemma for open partial differential relations, as well as explicit 1-jet deformations of metrics along the boundary.

This monograph develops an introduction to global analysis centered on the interaction between differential geometry, functional analysis, partial differential equations, and variational methods on Riemannian manifolds. Beginning with smooth and Riemannian geometry, it develops Sobolev spaces, distributions, interpolation and fractional regularity, differential and pseudodifferential operators on vector bundles, elliptic theory, heat methods, bounded geometry, and trace theorems. It then treats Fredholm and index theory, culminating in the Atiyah–Singer index theorem, followed by geometric evolution equations and Ricci flow, infinite-dimensional geometry on Banach and Hilbert manifolds, and variational methods including the direct method, Palais–Smale theory, deformation arguments, the mountain pass theorem, and the Nehari method. Particular emphasis is placed on explicit proofs, the passage from local Euclidean estimates to intrinsic global statements, and the precise geometric hypotheses required in compact, noncompact, and boundary settings. The text is intended for advanced undergraduate and graduate students, as well as readers approaching global analysis from geometry or differential equations.

We introduce an equivalence relation on exponential families, and prove that two exponential families are equivalent in this sense if and only if the corresponding induced Hessian manifolds are isomorphic. Moreover, we classify exponential families of order 1 with constant Hessian sectional curvature. To this end, we show that for an exponential family of order 1, it has constant Hessian sectional curvature if and only if the natural exponential family equivalent to the given family has a quadratic variance function (NEF-QVF). The classification coincides with that of NEF-QVF by Morris (1982) essentially.

Giuliano Basso asked whether every $n$-dimensional integral current space admits an integral current structure with the same characteristic set and multiplicity one almost everywhere. We give a negative answer already in dimension one. The obstruction is a parity phenomenon at odd-valence branch points. More precisely, if a one-dimensional integral current has unit multiplicity on the arms incident to an isolated vertex $v$, then the coefficient of its boundary at $v$ has the same parity as $\deg(v)$. Hence infinitely many odd-valence vertices force infinite boundary mass for every full-support unit-multiplicity current. We construct a compact geodesic, doubling, $1$-Ahlfors regular graph-like continuum $G$ of finite length and a boundaryless current $T\in\I_1(G)$ with $\set(T)=G$, while no $S\in\I_1(G)$ with $\set(S)=G$ can have multiplicity one $\Hh^1$-almost everywhere. We classify all full-support integral cycles on $G$. In particular, the least possible essential maximal multiplicity is exactly two, and the minimum mass of such a cycle is $3/2$. For finite metric graphs we identify a complementary sharp defect: the minimum boundary mass among unit-multiplicity currents equals the number of odd-degree vertices. Finally, by taking products with flat tori and using the precise Ambrosio–Kirchheim slice representation, we obtain boundaryless compact geodesic Ahlfors-regular counterexamples in every dimension.

We show that the fractional perimeter of any order $0<s<1$ is maximal, among all planar sets of fixed perimeter, for the disk.

We prove the Cartan-Hadamard conjecture (the sharp Euclidean isoperimetric inequality) in every dimension, assuming that the sectional curvature is pinched relative to a reference radius of curvature. The reference radius of curvature may vary, and the condition includes examples where the radius of curvature blows up, and the manifold has both unbounded (negative) and asymptotically vanishing sectional curvature. The key step in the proof is to control the entire convex hull of a quotient minimiser at its mean-curvature scale and compare the metric distortion with the surplus in the normalised hyperbolic isoperimetric profile at that same scale.

In this paper, we investigate Hamiltonian stationary Lagrangian currents with single-valued phases in open sets of Ricci-flat Kähler manifolds. We first establish the relation between phase and mean curvature for a class of integral Lagrangian currents in Ricci-flat Kähler manifolds. Then we derive several fundamental properties of integral Lagrangian currents with single-valued harmonic phases, including an almost monotonicity formula, an Allard-type regularity theorem, and the structure of limits of currents in this class. Finally, we apply the above results to study the regularity and rigidity of Hamiltonian stationary Lagrangian graphs in complex Euclidean space under certain conditions.

An old conjecture in Hermitian geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature is Kähler when the constant is nonzero and Chern flat when the constant is zero. Yang–Zheng introduced the real bisectional curvature as a generalization of the holomorphic sectional curvature and conjectured that, a compact Hermitian manifold with constant real bisectional curvature has zero constant and is Chern flat. They proved that the constant cannot be positive. In this paper, we first prove their conjecture. For a negative constant, we show that the metric is pluriclosed and use comparison with a negative Kähler–Einstein metric to obtain a contradiction. For the zero constant, we combine a torsion identity of Lin–Ren with the Bochner formula of Zhou–Zheng to prove Chern flatness. We also prove that a pluriclosed metric with constant negative holomorphic sectional curvature on a compact Kähler manifold is Kähler.

We prove quantitative instability results for the light ray transform in the Minkowski setting, and locally in the smooth and Gevrey categories. In the smooth Sobolev setting, we show that no modulus of continuity can be better than $t^\alpha$ for any $\alpha \in(0,1)$, while in the Gevrey setting we obtain an explicit logarithmic lower bound. On the other hand, using the relation between stationary geometries and magnetic(-potential) systems, we obtain a stability result for "time moments" of the light ray transform.

In this paper, we determine the nonpositive spectrum of the Yang-Mills Jacobi operator in Coulomb gauge at the standard $G_2$-instanton on $S^7$. We prove that the only negative eigenvalue is $-3$, with multiplicity $8$, and the kernel has dimension $20$. Thus the connection attains the sharp lower bound for the Morse index of nonflat Yang-Mills connections on $S^7$. We also give an explicit description of the kernal of the Jacobi operator, which contains Waldron's $15$-dimensional deformation space for the fixed nearly parallel $G_2$-structure. The 5 additional zero modes are isometry-induced variations that do not satisfy the corresponding linearized instanton equation.

In an earlier work, Yau and Zheng proposed a Hermitian analogue of a weak form of the Frankel conjecture, which states that any compact Hermitian manifold with positive Strominger-Bismut sectional curvature must be biholomorphic to the complex projective space. In this article, we confirm the conjecture in complex dimension two. This is achieved by a vanishing theorem for anti-self-dual harmonic 2-forms on such surfaces, utilizing a curvature decomposition formula by Ferreira and a refined Kato inequality.

We establish quantitative comparisons between the Laplace–Beltrami spectra of Kähler metrics of equal area on a compact Riemann surface. An estimate in terms of the oscillation of a potential gives bounds for fractional powers of reciprocal eigenvalues and explicit intervals for eigenvalue ratios. Bounds involving the gradient and Laplacian of the potential give further comparisons. On the Riemann sphere, we obtain estimates for individual eigenvalues, reciprocal sums and counting functions. For a Kähler metric on the unit 2-sphere which is centered in the sense that the unit normal vector field integrates to zero, the Dirichlet energy of the potential yields quantitative improvements of Hersch's bounds for the first positive eigenvalue and the sum of the first three reciprocal eigenvalues.

Let $X$ be a closed, smooth, orientable, positive definite $4$-manifold with $H_1(X ; \mathbb{Z}) = 0$. Let $p$ be a prime and suppose that $G = \mathbb{Z}_p$ acts smoothly on $X$ (if $p=2$ we also require an assumption on how $G$ acts on $H^2(X ; \mathbb{Z})$). Using equivariant Yang–Mills theory, Hambleton–Lee and Hambleton–Tanase proved (in the simply-connected case) that the fixed point set and tangential isotropy representations coincide with that of an equivariant connected sum of copies of $\mathbb{CP}^2$ on which $G$ acts linearly. We give a new proof of this result using equivariant Seiberg–Witten theory. Furthermore we also determine the weights of all equivariant line bundles on $X$, showing that these likewise coincide with that of an equivariant connected sum of linear actions on $\mathbb{CP}^2$.

Let $X:M^n\to\mathbb R^{n+1}$ be a complete, connected, two-sided minimal immersion without boundary, where $n=3,4$. We prove that finite $\delta$-index, finite Morse index, and finite total curvature are equivalent for $\delta>((n-1)/n)^2$, without assumptions on properness, volume growth, or topology. The main estimate shows that $\delta$-stability outside a compact set and finite-dimensional $H_c^1(M;\mathbb R)$ imply intrinsic Euclidean volume growth for $\delta>(n-2)/n$. We also deduce the sharp $\delta$-stable Bernstein theorem in this latter range from results of Hong–Li–Wang and Florit-Simon.

In this paper, we study h-Ricci-Bourguignon solitons on the product manifold D2 x R, where D2 is the Poincare disk equipped with its standard hyperbolic metric. We first establish that this manifold admits a gradient Ricci-Bourguignon h-soliton only if the non-zero function h depends solely on the manifold's last coordinate. Next, in the case where h:=1, we explicitly determine the vector field that makes D2 x R a Ricci-Bourguignon soliton. Finally, we provide the necessary and sufficient condition for the dual 1-form of this vector field to define a contact structure, which corresponds to the existence of an explicitly determined function independent of the manifold's second local coordinate, thereby allowing us to compute its Reeb vector field.

A minimal surface in $\mathbb{R}^3$ is $\textit{separable}$ if it is the zero set of $f(x)+g(y)+h(z)$, and $\textit{isotropic}$ if moreover $f=g=h$. We announce that there are exactly two non-planar isotropic separable minimal surfaces up to homothety and rigid motion, that they are conjugate, and that they are Schwarz's diamond surface $\mathrm{D}$ and Schwarz's primitive surface $\mathrm{P}$. In lattice-normalized coordinates their implicit equations are \[\mathrm{D}:\; \mathcal{D}(x)\mathcal{D}(y)\mathcal{D}(z)=1,\] \[\mathrm{P}:\; \mathcal{C}(x)+\mathcal{C}(y)+\mathcal{C}(z) =3\,\mathcal{C}(x)\mathcal{C}(y)\mathcal{C}(z),\] where $\mathcal{D}(t)=\frac{\operatorname{sn}\operatorname{dn}}{\operatorname{cn}}(K[\frac{1}{4}]t,\frac{1}{4})$ and $\mathcal{C}(t)=\operatorname{cn}(2K[\frac{3}{4}]t,\frac{3}{4})$. The first equation is classical – it is due to Schwarz, later Cayley, and appears in Nitsche's $\textit{Lectures}$ – we identify Nitsche's transcendental function exactly as $\operatorname{sn}\operatorname{dn}/\operatorname{cn}$ at parameter $\frac{1}{4}$. The second appears to be new; it is the symmetric member of the two-parameter family of Kim and Ogata (2024), and it proves their assertion that the family contains $\mathrm{P}$. Both surfaces are governed by one differential equation with two signs, \[\varphi'^{\,2}=1+2\cosh 2\varphi\ \ (\mathrm{D}),\] \[\varphi'^{\,2}=1+2\cos 2\varphi\ \ (\mathrm{P}),\] the two signs of the separation constant of the classical reduction; and in that common normalization the two lattice periods are $2K[\frac{1}{4}]$ and $2K[\frac{3}{4}]$, whose ratio is Schwarz's 1866 constant $K'[\frac{1}{4}]/K[\frac{1}{4}]=1.2792615...$, the necessary scale ratio for a conjugate pair. Proofs will appear in [D, P].

Let $M$ be a complex surface endowed with a holomorphic projective connection and let $C \subset M$ be a compact smooth holomorphic curve. Given two singular holomorphic foliations $\mathcal{F}$ and $\mathcal{G}$ near $C$, with $C$ not $\mathcal{G}$-invariant, we attach to every point $p \in C$ an index $\mathrm{Ind}(\mathcal{F},\mathcal{G},C,p) \in \mathbb{C}$ of $\mathcal{F}$ relative to the reference $\mathcal{G}$, and we prove that these indices add up to $T\mathcal{G} \cdot C$. When $C$ is $\mathcal{F}$-invariant the index is the Camacho–Sad index corrected by the order of tangency of $\mathcal{G}$ with $C$, and the index formula reduces to the Camacho–Sad formula. The main applications are two contact formulas. At a tangency point $p$ of $\mathcal{G}$ with $C$, let $k(\mathcal{G},C,p)$ be the ratio of the curvatures at $p$ of the leaf of $\mathcal{G}$ and of the curve; it is a projective invariant, although each curvature separately depends on a choice of metric. If $\mathcal{G}$ has no singular points on $C$ and only simple tangencies with $C$, then $$ \sum_{p} \frac{1}{1 - k(\mathcal{G},C,p)} = \frac{2}{3}\left( C \cdot C + g - 1 \right), $$ where $g$ is the genus of $C$: the number of tangencies depends on $\mathcal{G}$, but this weighted count does not. For a pencil of lines in the projective plane it is the Plücker formula for the class of a plane curve. The second formula asserts that, for a curve in general position with respect to $\mathcal{F}$ and $\mathcal{G}$ which is not a geodesic, the total index of the tangencies of $\mathcal{F}$ with $\mathcal{G}$ along $C$ equals the number of tangencies of $\mathcal{G}$ with $C$ minus one third of the number of inflection points of $C$; in particular, it does not depend on $\mathcal{F}$.

For every integer $n\ge2$, we construct a smooth complete Riemannian metric $G_n$ on $\mathbb{R}^n$ with full curvature norm at most one and injectivity radius at least one for which no $C^2$ isometric immersion into a finite-dimensional Euclidean space has bounded mean curvature. In dimension two, bounded second fundamental form would give uniformly controlled finite Jacobian representations of the Laplacian of the conformal factor. We construct disjoint conformal blocks for which the curvature remains bounded while the finite Jacobian representation cost tends to infinity. Taking Euclidean products gives all higher dimensions. The result answers Yau's Problem~52 negatively under the stronger assumption of bounded full curvature.

We introduce defect measures for the heat flow of a class of harmonic maps with free boundary and initiate their analysis. Our flow is motivated by singular perturbations on the boundary of the domain, emphasizing structural differences between harmonic maps with free boundary and their classical counterpart. The harmonic maps we consider are a reformulation of maps introduced by Da Lio and Rivière and arise in several problems in the spectral geometry of Steklov eigenvalues. Contrary to the classical case, the flow we consider exhibits features of global nature and the blow-up analysis turns out to be much more challenging.

We study the Gray-Hervella classes of invariant almost Hermitian structures on the maximal real flag manifold of type $\mathcal{A}_3$. We first obtain an explicit parametrization of all invariant almost Hermitian structures and establish necessary and sufficient conditions for an invariant almost Hermitian structure to belong to each Gray-Hervella class. For each nonempty class, we give an explicit parametrization of the corresponding set of structures, describe it as a smooth manifold and determine its dimension.

This paper is concerned with the study on Legendrian submanifolds with conformal Maslov form in the unit sphere $\mathbb{S}^{2n+1}$, which admits a Sasakian structure $(\varphi,\xi,\eta,g)$ for $n\ge2$. As the main result, we classify such submanifolds with constant sectional curvature, motivated by the classification result of the minimal Legendrian submanifolds with constant sectional curvature. Moreover, we prove that, for a closed Legendrian submanifold $M^n$ in $\mathbb{S}^{2n+1}$ with conformal Maslov form, if its sectional curvature satisfies the pinching $0\leq\sec_g\leq1$, then either $M^n$ is the totally geodesic Legendrian sphere with $\sec_g=1$, or $M^n$ is a closed embedded weighted Clifford torus with $\sec_g=0$. This extends the corresponding pinching theorem of Dillen–Vrancken (J Math Pures Appl 69:85–93 1990) from minimal Legendrian submanifolds to the conformal Maslov class under the same curvature bounds.

We study the diffusion and reaction effects of Bartnik's quasi-spherical equation to deform boundary mean curvature in fill-in problems with scalar curvature bounded below. The diffusion effect yields an explicit $L^p$-to-$L^\infty$ estimate for $\partial_tu=u^2\Delta u$, thereby extending the known upper bound for the minimum boundary mean curvature of fill-ins to a quantitative upper bound for its harmonic mean. For spin fill-ins, this bound is explicit and involves only coarse intrinsic boundary data. By introducing an absorbing reaction term, we also construct a deformation that transforms any nonnegative initial mean curvature into a terminal mean curvature with a uniform positive lower bound. For Gromov's conjecture on total mean curvature, this reduces the $H\geq 0$ case to Theorem A of Frenck, Hanke, and Hirsch \cite{FHH}, which assumes $H\geq\kappa>0$. This covers the case of non-spin boundaries, complementing their result for spin boundaries (Theorem B) for this conjecture.

We introduce and study a Yamabe-type variational problem associated with the curl operator on middle-degree forms on a closed oriented Riemannian manifold $(M^n,g)$ with $n\equiv3\pmod4$. The corresponding curl–Yamabe constant is defined by minimizing a conformally invariant, gauge-invariant quotient built from $\|{\rm curl}\,\alpha\|_{\frac{2n}{n+1}}$ and $\int_M\langle{\rm curl}\,\alpha,\alpha\rangle$, and its Euler–Lagrange equation is the critical curl–Yamabe equation ${\rm curl}\,\alpha=\frac{n+1}{2}|\alpha|^{\frac{2}{n-1}}\alpha$. Using sharp curl–Sobolev inequalities on the sphere and an Aubin-type cut-off construction, we prove that $0<Y_{\rm curl}(M,[g])\le Y_{\rm curl}(\mathbb{S}^n)$ for every such manifold, and that the infimum is attained whenever $Y_{\rm curl}(M,[g])<Y_{\rm curl}(\mathbb{S}^n)$. Moreover, for $n>3$ we establish the strict inequality for manifolds that are not locally conformally flat, via a test-form expansion detecting the Weyl tensor, and deduce existence of a minimizer solving the curl–Yamabe equation.

We study overdetermined equations on the hyperbolic and de Sitter planes by means of the support-function representation of zero mean curvature surfaces in Lorentz–Minkowski three-space. On the hyperbolic plane, solutions of $\Delta u-2u=0$ give rise to branched spacelike maximal surfaces. We prove that, on a bounded simply connected domain, nontrivial constant Dirichlet and Neumann data force the domain to be a geodesic disk, provided that the associated support quadric is nonlightlike; moreover, the solution is a multiple of the hyperbolic cosine of the distance from the center. On the de Sitter plane, the corresponding equation is the Klein–Gordon equation $\Box u+2u=0$, and its solutions generate timelike minimal surfaces wherever the support tensor is nondegenerate. We identify the support quadric and the constant-angle condition determined by constant Cauchy data, prove that local rigidity fails along every analytic noncharacteristic curve, and recover rotational symmetry from constant data on a global Cauchy circle. Thus the same support-function formalism reveals a sharp transition from elliptic rigidity to hyperbolic flexibility when the signature changes.

We prove that for every dimension $n \geq 2$ and parameter $p \in ]1,n[$, the hedgehog $u_0(x) = x/|x|$ is the unique minimiser of the $p$-energy $\mathcal{E}_p [u]:=\int_{\mathbf{B}^n}|\nabla u|^p\,{\rm d}x$ in the class of $W^{1,p}$-mappings from the $n$-dimensional unit ball $\mathbf{B}^n$ to the $(n-1)$-dimensional unit sphere $\mathbf{S}^{n-1}=\partial\mathbf{B}^n$ whose trace equal to the identity on $\mathbf{S}^{n-1}$. This answers in the affirmative an open question on p.29 in [Robert M. Hardt, Singularities of harmonic maps, Bull. Amer. Math. Soc. (N.S.) 34 (1997), 15–34].