Papers

The classical Heisenberg Pauli Weyl (HPW) inequality exhibits a strong rigidity phenomenon on Riemannian manifolds i.e. on Cartan Hadamard manifolds and those with non-negative Ricci curvature, the validity of the Euclidean HPW inequality or the existence of extremizers strictly forces the manifold to be isometric to Euclidean space, $\mathbb{R}^n$. This geometric discrepancy motivates the study of curvature dependent corrections and their associated stability properties. In this article, we investigate the geometric stability of the HPW inequality. Specifically, given a sequence of pointed Riemannian manifolds and appropriately normalized functions with a vanishing HPW deficit, we address whether the sequence converges to the corresponding model space in the pointed Gromov Hausdorff topology. We prove that for pinched Cartan Hadamard manifolds with sectional curvature bounded above by $c < 0$, the manifolds converge to the model hyperbolic space $\mathbb{H}^n_c$. In the non-negative Ricci curvature setting, we establish convergence to $\mathbb{R}^n$, up to a metric rescaling factor governed by the sequence's moment term. Consequently, we deduce that the known HPW inequality formulated via the asymptotic volume ratio (AVR) is suboptimal for non negatively Ricci curved manifolds not isometric to Euclidean space. To resolve this, we introduce a curvature corrected HPW inequality for this setting, analogous to the Cartan Hadamard case. Finally, we establish quantitative rigidity estimates in both curvature regimes, demonstrating that the HPW deficit when evaluated at Gaussian profiles which explicitly controls an appropriately defined distance to the respective model space.

We prove that any finite time collapsing Kähler Ricci flow on ruled surfaces develops a Type I singularity, such singularity is modeled on the standard product shrinker $\mathbb{P}^1\times \mathbb{C}$. As an application, we obtain the optimal collapse rate of fibers on ruled surfaces.

Let $X$ be a compact connected Riemann surface and $S\,\subset\, X$ a finite subset. We consider parabolic principal $G$–bundles $\mathcal{E}_{G}$ on $X$ with parabolic structure on $S$, where $G$ is a connected complex reductive affine algebraic group. Let $P\, \subset\, G$ be a parabolic subgroup and $\mathcal{E}_{P}\, \subset\, \mathcal{E}_{G}$ a reduction of structure group of $\mathcal{E}_{G}$ to $P$. We give a criterion for the existence of a parabolic Lie algebroid connection on $\mathcal{E}_{P}$ for any given parabolic Lie algebroid on $(X,\,S)$ whose anchor map is not surjective. More precisely, $\mathcal{E}_{P}$ admits a parabolic Lie algebroid connection if the reduction $\mathcal{E}_{P}\, \subset\, \mathcal{E}_{G}$ is parabolically infinitesimally rigid. In particular, the Harder–Narasimhan reduction of $\mathcal{E}_{G}$ admits a parabolic Lie algebroid connection.

We develop a theory of first order differential calculi in braided monoidal categories and classify braided covariant and bicovariant calculi on braided Hopf algebras. We show that, under certain conditions, bicovariant calculi can be transmuted to braided bicovariant calculi. For Radford–Majid biproducts, we combine bicovariant calculi on a Hopf algebra and braided bicovariant calculi on the corresponding braided Hopf algebra to covariant smash product calculi. The associated Maurer–Cartan form is shown to decompose into a direct sum of the Maurer–Cartan forms of the structure Hopf algebra of the quantum principal bundle and the braided Hopf algebra on the base. Geometrically, this construction realises the quantum affine extension of a given Hopf algebra, and we prove that the resulting quantum principal bundle is equipped with a frame resolution induced by the quantum Maurer–Cartan form. Building on this correspondence, we introduce and develop the notion of quantum $\textrm{G}$-structure, proving that quantum $\textrm{G}$-structures are quantum frame resolutions on the reduction. The theory is illustrated by examples based on transmutations of higher analogues of Sweedler's Hopf algebra and on the braided quantum plane, seen as a Yetter–Drinfeld module of $O_q(\mathrm{GL}_2)$.

For every pair of integers $3\leq k<n$ with $k$ odd, we construct a compact embedded rotational torus in $\mathbb{R}^{n+1}$ whose homothetic dilations satisfy the unnormalised $\sigma_k$-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has Hölder regularity $C^{1,1/k}$ and Sobolev regularity $W^{2,p}$ for every $1\leq p<k/(k-1)$. Away from two singular latitudes the torus is smooth; globally, the flow equation is interpreted using the weak shape operator of the associated Lipschitz boundary. Under rotational symmetry, the self-similar equation $\langle X,\nu\rangle=-\sigma_k$, where $X$ is the position vector and $\nu$ is the unit normal, reduces to a degenerate profile system. We solve this system by combining an odd-power desingularisation, a shooting argument, uniform radial and axial bounds, and a strict gap between the shooting parameters and the cylindrical radius. No classical $C^2$ rotational torus can satisfy the soliton equation, so the loss of regularity is unavoidable within the rotational toroidal class.

We establish codimension-independent mass bounds for geometrically confined area-minimizing rectifiable currents. In Euclidean space, we combine the confined-volume doubling theorem of Colding–Minicozzi with a current-theoretic squashing argument. This gives an affirmative answer to Lin's interior mass-bound problem for every algebraic projection multiplicity $Q$: the interior mass is bounded by $C(n)Q$, without an a priori mass bound at a larger scale. This result yields a degree-one Bernstein-type rigidity result under sublinear confinement. For the hyperbolic application, we make the curvature modification of the fixed-scale argument needed in a thin tubular neighborhood of a totally geodesic copy of $\mathbb{H}^n$. Combining this auxiliary estimate with a localized squashing estimate removes the doubly exponential local mass-growth condition from the boundary regularity results in [13].

Ricci curvature measures the average stability of geodesics under transverse perturbations, but how it should be defined in infinite dimensions is often unclear. This paper proposes a definition of Ricci curvature on the space of Hamiltonian diffeomorphisms on the two-dimensional flat torus $\mathrm{HDiff}(\mathbb{T}^2)$, the state space for ideal fluids. Our definition is based on Zeitlin's model, which approximates $\mathrm{HDiff}(\mathbb{T}^2)$ by finite-dimensional Lie groups $\mathrm{SU}(N)$. We derive a formula for the Ricci curvature tensor on $\mathrm{SU}(N)$ and provide numerical evidence for its convergence in the large-$N$ limit to our conjectured finite value. Additionally, we explore potential applications for hydrodynamics through the Lyapunov stability of gravest wave modes and Arnold's tradewind estimates for long-term weather predictability. Our framework extends to a wide range of settings. We demonstrate this by introducing Ricci curvature on the state spaces of fluids on rectangular domains, the Lagrangian averaged Euler equation induced by the $H^1$-Sobolev metric, and the quasi-geostrophic equation incorporating the Coriolis effect.

In the article, a novel framework for rough differential equations on finite-dimensional smooth manifolds driven by branched rough paths is developed utilizing the theory of natural bundles. The role of vector fields is played by sections of certain associated fiber bundles. The solutions are defined in a generalized Davie sense via a local approximation in such a way that they are invariant under changes of coordinates. Existence and uniqueness of the solutions is proved and a necessary and sufficient condition for the invariance of a submanifold for the solution is given. The approach allows the treatment of rough differential equations driven by fully branched rough paths on manifolds without directly relying on a shuffle product formula, bracket extension, or the Connes-Kreimer Hopf algebra and without imposing additional structure on the manifold.

Gromov conjectured that for a compact Riemannian manifold $X$ with boundary, the total mean curvature $\int_{\partial X} H$ is bounded above by a constant depending only on the intrinsic geometry of $\partial X$ and a lower bound on the scalar curvature of $X$. Previous results towards this conjecture require, in addition, a lower bound on the mean curvature of the boundary. In the present paper, we investigate whether this extra assumption is necessary. In dimension $2$, we show that no lower bound on the geodesic curvature is needed. We estimate the total geodesic curvature of the boundary in terms of its length and a lower bound for the Gauss curvature of the surface. This confirms Gromov's conjecture in $2$~dimensions without any extra assumptions. In contrast, we give examples showing that a lower bound on the mean curvature is genuinely needed in dimensions $n \ge 3$.

We prove that a compact Kähler manifold carrying a regular Riemannian foliation whose leaves are Lagrangian and minimal is flat. In order to prove this result, we establish a foliated version of an integral formula due to Ros.

In this paper, we study the asymptotic Plateau problem in hyperbolic space for hypersurfaces of constant $H_k$-curvature. We prove the existence of a smooth complete $k$-convex hypersurface in $\mathbb{H}^{n+1}$ satisfying \[ H_k(\kappa)=\sigma, \qquad \sigma\in(0,1), \] with prescribed asymptotic boundary at infinity. In particular, our result extends the range of the constant $\sigma$ in the existence theorem of Guan and Spruck [J. Eur. Math. Soc. 12 (2010), no. 3, 797–817] for $H_{k}$ curvature to the full interval $(0,1)$.

Consider a holomorphic vector bundle $E$ over a compact complex curve $C$ which decomposes as a sum of stable vector bundles. For the projectivization $\mathbb{P}(E)$, we prove that the existence of a compatible extremal almost Kähler (aK) metric of involutive type in the sense of Lejmi is equivalent to the existence of a Calabi extremal Kähler metric. This result rests on the Yau–Tian–Donaldson correspondence in terms of the moment polytope $\Delta$ for $\mathbb{P}(E)$, proved by the author and Yin in a previous work. The main advantage is that compatible extremal aK metrics of involutive type are solutions to a second-order linear PDE, rather than a fourth-order nonlinear PDE for Calabi's extremal Kähler metrics. As an application, we prove that when $E$ has rank $4$ and $C$ is an elliptic curve or the projective line, $\mathbb{P}(E)$ is a Calabi dream manifold, i.e. admits an extremal Kähler metric in every Kähler class.

In this article, we introduce a new distance function between hypersurfaces with free boundary. We show that our new quantity, which we call twisted Fermi distance, is monotone under mean curvature flow with free boundary. This overcomes the stumbling block that monotonicity of the usual distance function can fail for non-convex domains, and has several applications. Most importantly, we generalize the avoidance principle for free boundary Brakke flows, recently established by the first author for convex domains, to arbitrary domains. Using this, we then show that all results from our recent joint work, including the mean-convex neighborhood theorem and the uniqueness theorem for free boundary flows through cylindrical singularities, can be generalized to arbitrary domains without any convexity assumptions as well.

We work over a field of characteristic zero - primarily over an algebraic closure of the field of rational numbers. We prove a necessary and sufficient condition for a vector field whose coefficients are rational functions separating variables to be algebraically integrable, that is, the subring of rational functions killed by this vector field, its ring of first integrals, is of maximal possible dimension. We do it arithmetically by reducing the vector field modulo almost all primes. In particular, we verify the generalized Grothendieck–Katz p-curvature conjecture for foliations defined by these vector fields. Finally, we use the outcome of that verification to provide explicit formulas for coefficients of all algebraically integrable vector fields separating variables, and their first integrals.

We determine the Morse index and normal nullity of all constant-curvature minimal two-spheres in complex projective spaces, identify the normal Jacobi kernel as an $\SU(2)$-module, and prove that every normal Jacobi field is integrable. The formulas include all totally geodesic extensions. In a fixed ambient space, the normal nullity is constant along each Veronese sequence.

Triangulate a simple quadrilateral by connecting all vertices to an additional point. If the vertices carry values, the piecewise linear function can be assigned a Dirichlet energy. We show that the minimal Dirichlet energy as a function of the location of the inserted point is convex, and the location of the minimum is independent of the values at the corners - a quadrilateral has a discrete harmonic center, characterized by an equilibrium of currents across the inserted edges. It turns out that the fixed points of the Möbius involution swapping opposite corners of the quadrilateral are critical points of this energy, so the discrete harmonic center is Möbius-covariant. For tangential and cyclic quadrilaterals the center admits simple closed forms related to the circle centers. The center and its data-independence generalize to polytopes with d + 2 vertices in dimension d, but the conformal characterizations are special to four points in the plane.

In this paper, we prove that any complete, compact or noncompact, almost-Kähler gradient shrinking Ricci soliton is Kähler in arbitrary even dimension. Among other applications, combining our result with the classification of complete gradient shrinking Kähler-Ricci solitons in complex dimension two, we obtain a full classification of complete almost-Kähler gradient shrinking Ricci solitons in real dimension four.

Let $1 \leq q \le 2$ and denote by $2 \leq q'$ its corresponding conjugate exponent. We prove the continuity of solutions $u \in W^{1,(\frac{n}{n-1},q')}(B^n, \mathbb{R}^m)$ to the critical elliptic system $-\Delta u = \Omega \cdot \nabla u$ in dimension $n \ge 3$, where the potential $\Omega \in L^{(n,q)}(B^n, \mathfrak{so}(m) \otimes \wedge^1)$ is antisymmetric. First, we construct $P \in W^{1,(n,q)}(B^n, \mathrm{SO}(m))$ such that the PDE can be rewritten as $-\operatorname{div}(P^{-1}du) = \ast d\xi \cdot P^{-1}du$, which is nearly a Jacobian structure up to the rotation $P$. Second, we provide a Rivière's $\mathrm{GL}(m)$-Gauge in order to establish a "full" $(A,B)$-conservation law, i.e. $-\operatorname{div}(Adu)=d^\ast B \cdot du$. We show that the assumption on $\Omega \in L^{(n,q)} (B^n, \mathfrak{so}(m) \otimes \wedge^1) $ for $q\leq 2$ is optimal.

In this paper, we establish a stability estimate and a uniform estimate for the modulus of continuity of solutions to the degenerate complex Monge-Ampère equation in big cohomology classes. Consequently, we prove the uniqueness of solutions to complex Monge-Ampère mean field equations for a sufficiently small parameter.

There exists an adjoint pair (D, C) of functors between the category of diffeological spaces and that of topological spaces. In this article, by using the functors, we introduce two pairs of functors between the categories of sheaves on a diffeological space and on a topological space. Then, the adjointness of the novel functors up to the functors induced by the unit and counit of C and D is clarified. There is no natural functor between sites obtained by a diffeological space and a topological space. Therefore, the adjunctions on the categories of sheaves are elaborated without applying the general theory of topoi.

We study asymptotic properties of critical points of the two-dimensional vectorial Allen-Cahn energy with finitely many non-degenerate wells, subject to a homogeneous Neumann boundary condition. For any sequence with uniformly bounded energy, we prove that the limiting full and potential measures are supported on a closed countably 1-rectifiable set up to the boundary and satisfy the discrepancy relations. The potential measure defines a free-boundary stationary rectifiable varifold. Boundary mass may occur: weights are constant on regular boundary arcs, finite-type junctions obey projected balance, and the sole non-boundary branch meets the boundary orthogonally. This provides a Neumann-boundary extension of Bethuel's planar interior theory. The key idea is to read the boundary geometry from the limiting stress-energy tensor, which identifies the potential-energy measure as the stationary interfacial measure even in the presence of boundary concentration.

We study polygonal general helices in Euclidean three-space using the turning and signed torsion angles of the discrete Frenet frame. For helices whose axis is not orthogonal to the edge tangents, we prove that the global constant-angle condition is equivalent to the existence of a conserved Frenet-frame vector. On the generic branch, elimination of the auxiliary coefficient yields an intrinsic finite-step compatibility relation involving three consecutive turning angles and two consecutive torsion angles. Complementary phase and linear-subspace formulations cover the antipodal-binormal case. These characterizations reconstruct the helical axis and the helix angle and yield a sharp bound for each turning angle. We also give a spherical formulation through the tangent indicatrix: its vertices lie on a plane section of the unit sphere, which is a small circle in the non-orthogonal case, whereas the orthogonal case is exactly planar. A nonconstant Frenet-data example illustrates the criterion. Finally, for uniform chordal sampling of a smooth curve, the discrete Lancret-type quotient and the reconstructed helical direction converge with second-order accuracy.

Let $n\ge3$ and $D_t=\operatorname{diag}(t^{g_1},\ldots,t^{g_n})$ be a positive diagonal dilation family. We study connected embedded Euclidean hypersurfaces whose diagonal images are minimal. The level-set minimality operator splits into coefficients indexed by the pair sums $g_i+g_j$. Under pair-sum nonresonance, minimality at only $\binom n2$ distinct dilation parameters forces all pair coefficients to vanish. A dimension-reduction argument then shows, without any hypothesis on the coordinate components of the normal, that the second fundamental form vanishes identically. This yields an affine characterization. Repeated-weight helicoidal examples in every dimension and a resonant quadratic cone show that curvature cancellation can survive in genuinely nonuniform families. An application gives a finite-output-level rigidity criterion and an explicit representation for weighted-homogeneous production functions with minimal isoquants.

We study an exterior Bernoulli-type free boundary problem associated with the Riemannian $p$-Laplacian on a compact Riemannian manifold. Following the shape-optimization strategy of Ly and Seck, we formulate the corresponding volume-constrained shape optimization problem, derive its first shape derivative, obtain the overdetermined Bernoulli condition through a Lagrange multiplier, and establish a conditional monotonicity result under a Riemannian contraction hypothesis. The final successive-approximation argument will require additional geometric and boundary-flux stability assumptions.

We prove that the Higgs–Demailly system on a compact Riemann surface admits, for suitable fixed parameters, a smooth admissible solution at its terminal parameter if and only if the Higgs bundle is H-ample. Here H-ampleness is understood in the modified sense of Biswas–Misra–Ray. In particular, the argument gives an analytic proof of the equivalence between this H-ampleness and Griffiths-positive Hitchin–Simpson curvature. The proof extends the Demailly–Pingali–Murakami approach using a priori estimates and Leray–Schauder degree theory. The missing scalar lower bound follows from a Higgs-compatible quotient construction: a blow-up sequence produces a nonzero Higgs quotient of nonpositive degree, contradicting H-ampleness.

We prove uniqueness of finite-time geometric endpoints for the Möbius-invariant Willmore flow in $\mathbb{S}^3$ under uniform quantitative nonumbilicity. The multiplicity-counting varifolds converge, without reparametrization or Möbius renormalization, to a unique integral two-varifold. An intrinsic transport estimate gives quantitative total-variation convergence of the induced area measures on the fixed domain and bounded-Lipschitz Cauchy control of their pushforwards. Together with Allard compactness and rectifiability, this upgrades subsequential compactness to full-trajectory varifold convergence. The limit has generalized Euclidean mean curvature in $L^2$ with the natural endpoint lower-semicontinuity bound. For finite maximal trajectories with initial energy at most $8\pi$, Jakob's subsequential alternative becomes sequence independent: the limit is zero, or it has unit density and embedded Lipschitz support of genus zero or one. For Hopf-torus trajectories under the same energy bound, nonumbilicity is automatic and the anchored constant-speed profiles converge weakly in $W^{2,2}$ and strongly in $W^{1,2}$ and $C^{1,\alpha}$ for every $\alpha<\frac{1}{2}$. At infinite time, the same method yields a unique limit under an additional finite-dissipation-length condition.

We construct a compact, equiregular, ideal sub-Riemannian manifold of step $2$ such that, for every smooth positive measure, the $\mathrm{MCP}(K,N)$ fails for all $K\in\mathbb{R}$ and $N \in (1,\infty)$. This shows that the real-analyticity assumption in the measure contraction theorem of Badreddine and Rifford in arXiv:1712.09900v2 cannot be replaced by smoothness. Moreover, our structure is ideal, i.e. it admits no non-trivial abnormal minimizing geodesics. Although failures of the measure contraction property for ideal structures were recently obtained in the higher-step setting, our construction shows that the phenomenon can already occur on compact, equiregular, ideal structures of step $2$. The proof exploits a differential consequence of the measure contraction property, namely a uniform upper bound for the sub-Laplacian of the squared distance near its base point, and constructs a structure for which this quantity is unbounded.

We consider isometric $\mathbb{Z}_{p}$-torus actions on closed, positively curved manifolds, obtaining a homotopy classification result. This allows us to strengthen on the improvement of the $3n/8$ half-maximal symmetry rank result of Fang-Rong and Ghazawneh given in Theorem B and Corollary C in the work of the author and Searle. Along the way, we generalize a sphere-embedding construction of Wilking from the binary case to arbitrary primes in Theorem C, which is a geometric construction of independent interest.

For an endomorphism $\phi$ of a closed Riemannian manifold $M$, we study the pushforward action $\phi_\ast$ on the Wasserstein space $\mathcal{P}(M)$ at a measure $\mu_0$ preserved by $\phi$. We show that if $\phi$ is a $C^2$ covering map and $\mu_0$ has positive $C^1$ density, then $\phi_\ast$ is Gâteaux differentiable at $\mu_0$ along tangent directions, with derivative given by the transfer operator of $\phi$ acting on vector fields, followed by orthogonal projection onto the tangent space. The derivative is the adjoint of the Koopman operator restricted to the tangent space, and its fixed space consists of the directions in which $\mu_0$ can be deformed while preserving invariance to first order. For appropriate pairs of endomorphisms of $\mathbb{T}^d$, we compute the intersection of their fixed spaces, show that it contains an infinite family of linearly independent continuous vector fields, and construct, for every $n$, an embedded $n$-dimensional family of measures that are nearly invariant under both endomorphisms. First-order rigidity therefore fails in every dimension. In particular, rigidity phenomena such as higher-dimensional analogues of the Furstenberg conjecture, if true, are genuinely nonlinear.

With any holomorphic vector bundle $E$ over a compact base one can associate a line bundle, denoted $O_{PE}(1)$. According to a conjecture of Griffiths, if $O_{PE}(1)$ admits a positively curved hermitian metric $k$, then $E$ also admits a positively curved hermitian metric, $h$. In this paper we show that, while the conjecture may be correct, it is not possible to obtain $h$ out of $k$ by a fiberwise construction that is functorial.