Neckpinch — Eigenvalues & spectrum

Blowdowns along Ricci flow in dimension four

Thu, 29 Oct 2026 00:00:00 GMT

Dain Kim, Tristan Ozuch — We show that the blowdown shrinking Ricci soliton of Feldman, Ilmanen and Knopf is strictly dynamically stable in the following strong local sense: (1) The Ricci flow from every metric on a closed 4-manifold containing a region close to a rescaling of a sufficiently large compact subset of the soliton in either orientation and with controlled geometry elsewhere develops a singularity with unique tangent flow modeled on the soliton. (2) The singular time slice has an isolated conical singularity, and by Gianniotis-Schulze the flow can be continued by blowing down a sphere of self-intersection ±1. (3) On a closed 4-manifold, the set of metrics whose Ricci flows have their first singularity modeled on the soliton is C¹⁴-open. It is nonempty if and only if the manifold contains a sphere of self-intersection ±1. (4) The soliton is strongly rigid among shrinking Ricci solitons. As applications, we construct C¹⁴-open sets of initial data whose Ricci flows perform any prescribed ordered sequence of blowdowns. (1) Every closed simply connected smooth 4-manifold, including spin and exotic examples, can be reached by Ricci flow from a standard kℂP}²\#ℓℂP}², through finitely many blowdown singularities. (2) On the standard 2ℂP}²\#9ℂP}², there are infinitely many open sets of initial metrics whose Ricci flows, after a single blowdown, reach pairwise nondiffeomorphic manifolds. The proof uses the spectral analysis of Naff and the second author and deals with the gauge-fixing problem along Ricci flow through a Ważewski argument inspired by Stolarski. We also use Bamler-Lai's PDE-ODI principle. Except for continuation (requiring nonnegative curvature of the asymptotic cone), all results apply to linearly stable nonflat asymptotically conical shrinkers in all dimensions.

Ricci flow of Integrable Sigma Models, Hitchin Systems and Quantum Seiberg-Witten theory

Thu, 22 Oct 2026 00:00:00 GMT

G. Bonelli, S. Lacroix, T. Pedroni, A. Tanzini — We construct a family of integrable 2d sigma models through 4d Chern-Simons theory and show that their classical point-particle reduction corresponds to a Hitchin system. The integrable Hitchin quantisation then provides a natural candidate Hamiltonian for the one-loop reduced dynamics of the sigma models, which coincides with the Perelman operator in all cases examined. This allows us to study the corresponding spectral problems using 4d 𝓝=2 gauge theory and 2d conformal field theory methods. For the sausage model, we derive exact quantisation conditions and compute exponentially small corrections essential for the agreement with numerical finite-volume results.

Finite-Shift Inverse-Concavity and Exterior Noncollapsing for Fully Nonlinear Curvature Flows of Nonconvex Hypersurfaces

Wed, 21 Oct 2026 00:00:00 GMT

Weimin Sheng, Ye Zhu — We give a finite negative-shift criterion for preserving a prescribed exterior noncollapsing bound under fully nonlinear curvature flows whose hypersurfaces need not be convex. The criterion uses the optimized two-point expression of Andrews and Langford. We verify it by a uniform global two-matrix estimate for the explicit speed F_ε(A)= tr A- εfracnormA^(∘)² tr A on the open cone Γ=A∈Sym(n): tr A>0, normA^(∘)< tr A\. Here A^∘ denotes the trace-free part of A. This cone contains the positive cone and also matrices with a negative eigenvalue, while the displayed speed is concave and non-convex. For an explicit range of parameters depending on the initial exterior bound, the cone and that bound are preserved throughout the smooth embedded existence interval. Together with the known interior estimate for concave speeds, this gives two-sided tangent-ball bounds for this nonlinear family, including closed nonconvex initial hypersurfaces.

Asymptotically conical shrinking ends of the r-mean curvature flow: rotational construction and uniqueness

Fri, 09 Oct 2026 00:00:00 GMT

Hilário Alencar, Márcio Batista, Gregório Silva Neto — For 1≤ r≤ n-1 we study shrinking self-similar ends of the r-mean curvature flow, Sᵣ=c⟨ X,N⟩ with c<0, that are asymptotic to a cone. For every rotational cone of positive slope we construct a smooth embedded rotational end and prove that its profile is the only positive solution of the profile equation with the prescribed slope at infinity. Its angular defect xu'−u is strictly negative, which yields r-admissibility, and the profile has a complete asymptotic expansion for u(x)/x in powers of x^(−(r+1)), valid after arbitrary differentiation. The profiles depend real analytically on the slope and, on sufficiently distant exterior regions, foliate the image of the corresponding family. Our main result removes the symmetry assumption: two ends that converge smoothly, with multiplicity one, to the same circular cone coincide outside a compact set. The proof transforms the equation for their difference into an almost-eigenfunction inequality for a drift Laplacian on an auxiliary conical metric and applies Bernstein's asymptotic analysis on weakly conical ends. The argument extends to strictly r-admissible cones whose links have constant principal curvatures.

Complex Monge-Ampère equations on Hermitian Manifolds: From bounded to smooth solutions

Mon, 05 Oct 2026 00:00:00 GMT

Papa Badiane, Chinh H. Lu, Ahmed Zeriahi — We study regularity of bounded weak solutions to complex Monge-Ampère equations on compact Hermitian manifolds with right-hand side possibly decreasing in the unknown. Our proof relies on the domination principle and a priori estimates, partially motivated by the Monge-Ampère eigenvalue problem. Our main result roughly says that any bounded solution is smooth in the regular locus of the data. When the right-hand side is strictly positive and smooth, we recover known results by Nie, Ko lodziej-Nguyen for the Hermitian case and Székelyhidi-Tosatti for the Kähler case, which rely on the regularizing property of the Kähler-Ricci flow.

Conformal Killing–Yano Ricci solitons: Structure, compatibility, and rigidity

Sun, 20 Sep 2026 00:00:00 GMT

Mohammadjavad Habibivostakolaei, Abbas M. Sherif, Yen-Kheng Lim — 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.

Harmonicity and Existence of Algebraic Generalized Ricci Solitons

Thu, 10 Sep 2026 00:00:00 GMT

Huibin Chen, Zhiqi Chen, Fuhai Zhu — In this paper, we study the harmonicity and existence of algebraic generalized Ricci solitons. Firstly, we characterize harmonic torsion of algebraic generalized Ricci solitons on arbitrary metric Lie algebras by an identity involving the Killing form. In particular, positive semidefiniteness of the Killing form implies harmonicity without a unimodularity assumption. Then, we construct generalized nilsolitons with nonzero torsion on indecomposable three-step nilpotent Lie algebras admitting no classical nilsoliton, in dimension seven and in dimensions 6m+d for m>d≥1. Furthermore, we provide a spectral obstruction for nilpotent Lie algebras with abelian derived algebra and an obstruction based on the action of derivations on the quotient by the center. Combining these obstructions with explicit constructions, we prove that the filiform Lie algebra mathfrak m₂(n), n≥5, admits a generalized nilsoliton if and only if 5≤ n≤8.

The Weyl Law Meets Large-Scale Regularity on Ricci Shrinkers

Tue, 08 Sep 2026 00:00:00 GMT

Junrong Yan — We prove that the weighted Laplacian, or equivalently its conjugate Schrödinger operator, on every complete gradient Ricci shrinker satisfies the classical Weyl law. The main difficulty is that uniform bounded geometry is not known for general Ricci shrinkers. To overcome this, we establish a large-scale regularity property for complete gradient Ricci shrinkers and apply it to the spectral asymptotics of the weighted Laplacian. We prove that, inside large geodesic balls of radius R, the region where the curvature radius is smaller than R⁻¹ occupies an asymptotically negligible proportion of the volume. The proof uses the Ricci flow associated with the shrinker, together with the curvature-radius estimates and Sobolev inequalities of Li–Wang.

Gradient Ricci Solitons on PTR-Submanifolds of Complex Space Forms

Sat, 05 Sep 2026 00:00:00 GMT

Jae Won Lee — We study gradient Ricci solitons on partially totally real (PTR) submanifolds of complex space forms. In nonzero ambient holomorphic sectional curvature, the Gauss–soliton equation reconstructs the square P² of the tangential Kähler operator from the Hessian endomorphism and an extrinsic Ricci operator. Consequently, the PTR tangency spectrum is encoded by intrinsic soliton data together with the second fundamental form. For compact minimal PTR solitons with positive ambient curvature we obtain λ≤ c(n-1)/4, with a strict version under a relative-nullity condition. We also prove compact triviality criteria, an exact trace-free anisotropy identity, a parallel-data rigidity result, and a nonexistence theorem for compact real hypersurfaces. A flat multi-level Gaussian model and a nonflat Segre CR-product provide explicit consistency checks and identify the limiting cases of the reconstruction formulas.

Positive τ-bi-Ricci curvature and Mean curvature flow with surgery in hyperbolic space

Wed, 26 Aug 2026 00:00:00 GMT

Tianci Luo, Yong Wei, Rong Zhou — Let n≥3 and 0≤τ≤2. We prove that every smooth closed connected immersed hypersurface in hyperbolic space whose induced metric has positive τ-bi-Ricci curvature admits a mean curvature flow with surgery which has only finitely many surgery times and terminates. In the range compatible with cylindrical necks, the key ingredients are a preserved quantitative spectral pinching condition, which converts the intrinsic hypothesis into uniform two-convexity, and cylindrical and derivative estimates that remain valid across the hyperbolic standard neck replacement. At the endpoint n=3, τ=2, positive τ-bi-Ricci curvature is positive Ricci curvature and forces strict convexity, so the ordinary mean curvature flow converges to a round point. Consequently, the underlying manifold is diffeomorphic to a sphere or to a finite connected sum of copies of Sⁿ⁻¹×S¹. If the initial hypersurface is embedded and bounds a compact domain, that domain is a one-handlebody, namely a ball with finitely many one-handles attached.

Singularity formation and self-shrinkers for the parabolic Bernoulli problem

Sun, 23 Aug 2026 00:00:00 GMT

Hunter Liu, Sebastian Munoz, Zihui Zhao — We study singularity formation for classical solutions of the parabolic Bernoulli free boundary problem and the associated self-similar profiles. Inspired by Ricci and mean curvature flows, we introduce a Type I assumption on the blow-up rate at a singular free boundary point, under which the parabolic rescalings are compact and every tangent flow is a non-trivial self-shrinking solution of the elliptic profile equation. The proof establishes convergence of the positivity sets and recovers the Bernoulli condition in the limit, despite the absence of a minimizing structure. In the subcritical regime, every tangent flow is a double plane. We next classify all radial self-shrinkers. Besides the known ball and exterior profiles, there is a unique annular profile. We obtain dimension-uniform bounds and sharp asymptotics for its inner and outer radii, including a quantitative outward bias of its midpoint. These delicate estimates yield a complete classification of the signs of the linear spectrum for the annular solution in every dimension. As a consequence, the ball is dynamically stable modulo ambient symmetries, whereas the annulus is dynamically unstable, with genuine unstable modes of angular degrees 0,1,2,3. We finally prove, from the spectral nondegeneracy of the compact radial profiles, that whenever one tangent flow is a ball or an annular solution, the tangent flow is unique. One borderline spectral sign in dimensions 2≤ n ≤ 4000 is verified by a rigorous computer-assisted argument using interval arithmetic.

The Fisher Metric of the Ricci Flow Heat Kernel

Fri, 14 Aug 2026 00:00:00 GMT

Bennett Chow, Robert Koirala — We introduce and study a Fisher information metric g^F_τ associated to the conjugate heat kernel of a Ricci flow (Mⁿ,gₜ). This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that g^F_τ is monotone in scale and satisfies g^F_τ≤ gₜ. We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect gₜ−g^F_τ. This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities 0

Weinstock inequalities for outward-minimizing domains

Wed, 12 Aug 2026 00:00:00 GMT

Chaoqun Gao, Yong Wei, Rong Zhou — We prove sharp Weinstock inequalities for the first nonzero Steklov eigenvalue of smooth outward-minimizing domains in Euclidean space and hyperbolic space. The method is based on the weak inverse mean curvature flow of Huisken–Ilmanen. The main new ingredient is an endpoint distributional monotonicity argument obtained from the calibrated weak formulation and the Gauss–Green formula for divergence-measure fields.

Open full-metric formation and local marked first-order asymptotic moduli of FIK blowdown singularities

Sun, 09 Aug 2026 00:00:00 GMT

Henry Shin — We prove that the Feldman–Ilmanen–Knopf (FIK) blowdown singularity has a nonempty open nonlinear formation basin in the full space of smooth Riemannian metrics and, on a smaller neighborhood, local marked first-order asymptotic moduli. Given a closed connected oriented Riemannian four-manifold, an implantation point, and a positive time bound, its oriented blow-up admits a relatively C^(2,α)-open set of smooth metrics whose Ricci flows form, before that time, a localized FIK singularity with global Type-I curvature control. No symmetry, Kähler condition, or finite-dimensional tuning is imposed. Curvature stays uniformly bounded outside the implantation region; fixed-convention marked rescalings converge along the full singular-time sequence to the ancient FIK flow; and the exceptional sphere has sharp FIK asymptotics for area, intrinsic diameter, and curvature. To our knowledge, this is the first open full-metric formation theorem for a finite-time singularity modeled on a noncompact, noncylindrical shrinker. A self-contained spectral certification identifies the weighted FIK operator's nine-dimensional nonnegative space with the scaling and diffeomorphism directions, and exact modulation removes this block. On a smaller little-Hölder h^(2,α) neighborhood, a fixed positive-time restart yields the marked first-profile coordinate A₁=λ_∞^(−γ₁)V_∞∈ E₁. This amplitude is a split C¹ submersion and locally the projection onto E₁. Equality of amplitudes characterizes marked first-order agreement; a transverse disk uniquely realizes each sufficiently small amplitude; normalized two-flow differences converge to the corresponding Jacobi field; and the amplitude determines the first quadratic scale and phase response. This gives a complete local marked first-order profile coordinate for FIK blowdown.

Conformal-Mapping Method for Horizon Multipoles in Numerical Relativity: Implementation, Kerr Validation, and Applications Beyond Axisymmetry

Sat, 08 Aug 2026 00:00:00 GMT

Yeong-Bok Bae, Young-Hwan Hyun, Gungwon Kang — We present a numerical method that constructs geometrically defined coordinates on black-hole horizons, from which the multipole moments are computed without assuming axisymmetry. This method, which we denote the conformal-mapping method (CMM), provides a numerical realization of the conformal construction proposed by Ashtekar et al. in 2022, combining discrete Ricci flow, spectral embedding onto the unit sphere, and Möbius gauge fixing by the vanishing-area-dipole condition. We first test the CMM against analytic Kerr benchmarks, and then apply it to an equal-mass, non-spinning binary black-hole merger. We also compare it with an approximate-symmetry-based method. The CMM allows the multipole moments to be expressed in a fixed reference frame, whereas the symmetry-adapted frame can reorient abruptly when the preferred approximate axis changes. In a frame aligned with the orbital angular momentum, the amplitude of the quadrupole mode grows during inspiral and decays after merger, displaying a qualitative ringdown behavior. These results show that the CMM is a useful tool for studying horizon geometry in dynamical situations where no stable symmetry axis is available.

Contracting Transport Maps on Riemannian Manifolds

Thu, 30 Jul 2026 00:00:00 GMT

Shrey Aryan, Bang-Xian Han, Zhuo-Nan Zhu — We use inverse mean curvature flow to construct bi-Lipschitz maps that preserve normalized volume and decrease distances. These maps send a round sphere onto any smooth closed strictly convex hypersurface in a sphere and a flat disk onto any smooth strictly convex free-boundary disk in a Euclidean ball. In dimension two, this proves a conjecture of E. Milman for every smooth two-sphere with Gaussian curvature at least one and gives an analogous result for nonnegatively curved disks whose boundary has geodesic curvature one. The spherical result proves the two-dimensional case of the spectral comparison conjectured by Colding and Minicozzi. Counterexamples in dimensions n≥3 show that the restriction to dimension two is sharp. Furthermore, an equivariant extension of this construction yields, for every n≥2, a contracting transport map from the uniform probability measure on a round hemisphere to the uniform probability measure on any closed geodesically convex subset of positive volume. This settles the remaining uniform-target case of a question raised by Beck and Jerison. In dimension two, we also find geometric conditions under which the uniform measure on the hemisphere can be transported by a contracting map to a broad class of nonuniform probability measures supported on domains in a hemisphere.

Spectral bounds for f-Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers

Fri, 24 Jul 2026 00:00:00 GMT

Fei He — We study the spectrum of the f-Laplacian on complete gradient Ricci shrinkers. Upper and lower bounds for the k-th eigenvalue are established in terms of the volume growth rate. Both bounds are shown to be sharp in the exponent. The method extends to f-Laplace-type operators on vector bundles; as an application we obtain explicit upper bounds for the Betti numbers.

The Penrose conjecture for initial data sets satisfying a 2-convexity condition

Wed, 22 Jul 2026 00:00:00 GMT

Conghan Dong — Let (M³, g, k) be a smooth, connected, asymptotically flat initial data set with connected outermost past apparent horizon Σ. We prove the Penrose conjecture, namely that m_ADM(g) ≥ √(|Σ|)/(16 π), under the assumptions of the dominant energy condition and the 2-convexity condition that the sum of the two smallest eigenvalues of k is nonnegative. The main tool is the σ-inverse mean curvature flow, together with a monotonicity formula developed in.

The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow

Sun, 05 Jul 2026 00:00:00 GMT

Fagui Li, Yuhang Zhao — In this paper, we prove a spectral upper-pinching theorem for complete properly immersed self-shrinking hypersurfaces. Our argument is inspired by the second author's recent work. If λ_ρ(Σ)≥λ>0 and S=|A|²<1+λ, then Σ is either a hyperplane or a generalized round cylinder. In the properly embedded case, the Ding–Xin and Brendle–Tsiamis weighted Poincaré estimate gives λ_ρ(Σ)≥1/2. Consequently, the pointwise upper pinching S<3/2 forces Σ to be a hyperplane or a generalized round cylinder. For embedded self-shrinking surfaces in ℝ³, we also obtain the endpoint case S≤3/2. These results remove the lower pointwise pinching assumption in the corresponding embedded upper-pinching range and improve the ranges in earlier work of Ding–Xin , Cheng–Wei , and Lei–Xu–Xu .

Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamics

Fri, 03 Jul 2026 00:00:00 GMT

Maxwell Stolarski, Wei-Bo Su — For each integer K≥2 when n≥4, and for K=2,3,4 when n=3, we construct an almost-calibrated Lagrangian mean curvature flow L_K(t) in ℂⁿ, starting from initial data arbitrarily close to being special Lagrangian, which develops a finite-time Type II singularity at time T with the explicit curvature blow up rate sup_L_K(t) |A_L_K(t)| ∼ (T-t)^(−K/2) as t↗ T . The tangent flow at the singularity is a transverse pair of cohomogeneity-one special Lagrangian cones, while the Type II blow-up limit is a smooth cohomogeneity-one special Lagrangian desingularization. This gives a quantitative construction of Type II blow-up for a fully nonlinear parabolic PDE arising from cohomogeneity-one Lagrangian mean curvature flow. Our construction is based on a modulation analysis around a shrinking family of cohomogeneity-one special Lagrangian desingularizations, using the perturbative spectral theory developed in the companion paper.

Spectral Analysis for Finite-Time Singularities of Lagrangian Mean Curvature Flow

Fri, 19 Jun 2026 00:00:00 GMT

Maxwell Stolarski, Wei-Bo Su — Let 𝓒 be a G-invariant special Lagrangian cone admitting a scaled family of G-invariant special Lagrangian desingularizations a overline L which converge to 𝓒 as a↘ 0. We study the linearized self-shrinker operator on aoverline L in a Gaussian weighted L² space of G-equivariant functions. For 0

Ricci flow for the Bures–Helstrom qubit metric

Wed, 17 Jun 2026 00:00:00 GMT

Andrew Lesniewski — The Bures–Helstrom metric is the minimal monotone Riemannian metric on the state space of a qubit. With the quantum Fisher normalization used here, it identifies the Bloch ball with a geodesic hemisphere of the unit round three–sphere. We describe its Ricci flow explicitly. In a general rotationally symmetric gauge the flow is a coupled system for the radial lapse and warping factor; a single scalar equation appears only after a Hamilton–DeTurck gauge choice. In the corresponding moving DeTurck frame the squared warping function Ψ=Φ² satisfies the linear forced heat equation \beginequation* D_tΨ=Ψ_ss-2, \endequation* while the fixed-lapse coordinate form contains the associated transport term. Since the Bures–Helstrom metric is Einstein, the geometric flow itself is the homothetic shrinker \beginequation* g(t)=(1-4t)g_BH, \endequation* with scalar curvature 6/(1-4t) and extinction time T=1/4. Thus the metric remains inside the monotone cone for all t

Spectral Properties of the Chen-Nagano Gauge on Einstein Manifolds

Mon, 25 May 2026 00:00:00 GMT

Sergey Stepanov — The stability and deformation theory of Einstein metrics traditionally relies on the classical Berger-Ebin transverse-traceless gauge, which structurally decouples the scalar trace from the divergence-free component of metric perturbations. In the present paper, we introduce a new spectral-geometric framework based on the Chen-Nagano gauge condition. This condition naturally arises from the harmonicity of the identity map and is intrinsically satisfied by the Ricci tensor itself via the contracted second Bianchi identity. Unlike the classical transverse-traceless framework, the Chen-Nagano gauge preserves a nontrivial interaction between the trace and trace-free sectors of a deformation. We establish a first-order differential relation proving that the divergence of the trace-free part is completely governed by the gradient of the scalar trace. Utilizing commutation formulas on Einstein manifolds, we derive a second-order spectral coupling relation that links the Lichnerowicz Laplacian to a shifted scalar operator. As a primary geometric consequence, we prove that under suitable spectral pinching assumptions, the Chen-Nagano gauge collapses to the classical transverse-traceless gauge. Specifically, we show that on compact connected negatively curved Einstein manifolds, any volume-preserving Chen-Nagano harmonic deformation whose trace-free component lies below a specific spectral threshold determined by the Einstein constant is necessarily transverse-traceless. Furthermore, we connect this rigidity to the curvature operator of the second kind, establishing explicit lower spectral bounds. Finally, we provide a dynamical interpretation within the Ricci flow framework, demonstrating that the linearized Ricci flow under the Chen-Nagano gauge reduces to a strictly parabolic equation governed by the Lichnerowicz Laplacian, ensuring exponential decay of admissible perturbations.

Spectral Obstructions to Contracting Transport Maps on Curved Spaces

Sat, 23 May 2026 00:00:00 GMT

Shrey Aryan — Caffarelli's contraction theorem states that the Brenier optimal transport map from the standard Gaussian measure to a more log-concave probability measure is 1-Lipschitz. Motivated by this theorem, Milman [Mil18] formulated several conjectures for the round sphere and for weighted manifolds satisfying the curvature-dimension condition CD(ρ,∞). Recently, Beck and Jerison [BJ21] raised related questions on the round hemisphere. The existence of a contracting transport map implies a corresponding spectral comparison. In the spherical setting, this comparison was also conjectured by Colding and Minicozzi [CM98] for compact manifolds with Ricci curvature lower bounds. In this work, we construct counterexamples to the corresponding spectral comparisons on spheres and on weighted manifolds satisfying CD(1,∞) in dimensions d≥4, yielding obstructions to contracting transport maps. In dimensions d≥5, the weighted counterexamples can be chosen to satisfy Ric_g≥ 0 and ∇_g² V ≥ g separately. In dimension two, we use inverse mean curvature flow to construct a contracting transport map from the suitably rescaled round sphere to every closed connected Riemannian surface satisfying the same positive Ricci curvature lower bound. This implies the spectral comparison in dimension two. Together with the recent counterexample in dimension three by Lin, Wang, and Xu [LWX26], this settles the spherical spectral comparison in every dimension d≥2. Using the same method, we also construct a contracting transport from the uniform probability measure on a hemisphere onto the normalized uniform measure on any geodesically convex subset of positive volume, thereby answering affirmatively the remaining case of a conjecture by Beck and Jerison [BJ21] following the work of Fathi, Fradelizi, Gozlan, and Zugmeyer [FFGZ26].

Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers

Fri, 22 May 2026 00:00:00 GMT

Xu Cheng, Franciele Conrado, Neilha Pinheiro, Detang Zhou — Let (M, g, f, τ) be a complete Ricci shrinker satisfying Ric+∇²f=g/2τ and let R denote its scalar curvature. For a confined function V on M, we obtain a lower bound for the lowest eigenvalue of the Schrödinger operator −Δ+R/4+V, expressed in terms of an integral quantity involving V and the shrinker entropy, and the equality case is characterized by the potential functions. We further generalize this estimate to complete Riemannian manifolds via Perelman's μ-functional. We also study the drifted Schrödinger operator −Δ_f+V on smooth metric measure spaces. In particular, on Ricci shrinkers, we derive a lower bound for its lowest eigenvalue, with equality if and only if V is affine.

Strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow

Sat, 16 May 2026 00:00:00 GMT

Hanbing Fang, Yu Li — In this paper, we extend the results of to generalized cylinders. More precisely, we establish a Lojasiewicz inequality for the pointed 𝓦-entropy in Ricci flow under the assumption that the geometry near the base point is close to a generalized cylinder ℝ^k × N^(n-k), where N is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition. As an application, we prove the strong uniqueness of generalized cylindrical tangent flows. Furthermore, we show that the subset 𝓢^k_qc(N)⊂ 𝓢^k, consisting of points at which some tangent flow is given by ℝ^k × N^(n-k) or its quotient, is horizontally parabolic k-rectifiable.

Quantitative Spectral Stability for an Embedded Annulus under Coupled Curve Shortening and 2D Ricci Flows

Wed, 21 Jan 2026 00:00:00 GMT

Mohammadjavad Habibivostakolaei — We study the spectral stability of Dirichlet eigenvalues on an embedded annulus whose boundary evolves by curve shortening flow while the ambient surface evolves under the two dimensional Ricci flow using variational formulas, Rellich–type identities, and harmonic capacity methods, we relate eigenvalue variations to geometric deficit and modulus. We establish quantitative bounds comparing the spectrum of the evolving annulus with that of a flat cylinder of equal modulus. As a consequence, we obtain geometric stability and a spectral gap estimate controlled by the deficit functional.

Classification of ancient ovals in higher dimensional mean curvature flow

Wed, 14 Jan 2026 00:00:00 GMT

Beomjun Choi, Wenkui Du, Ziyi Zhao — We study compact non-selfsimilar ancient noncollapsed solutions to the mean curvature flow in ℝⁿ⁺¹, called ancient ovals. Our main result is the classification of k-ovals: any k-oval (characterized by having cylindrical blow down ℝ^k× S^(n-k) and the quadratic bending asymptotics) belongs, up to space-time rigid motions and parabolic dilations, to the family of ancient ovals constructed by Haslhofer and the second author. Assuming the nonexistence of exotic ovals (recently proved by Bamler-Lai), this yields a classification of all ancient ovals and identifies the moduli space, modulo symmetries, with an open (k-1)-simplex modulo the symmetry of simplex. Although these conclusions are contained in the recent breakthrough of Bamler-Lai classifying all ancient asymptotically cylindrical flows and resolving the mean convex neighborhood conjecture, we give an alternative argument for the independently obtained classification of k-ovals in arbitrary dimensions based on a different spectral parametrization.

A Dynamical Approach to the Berezin-Li-Yau Inequality

Mon, 01 Dec 2025 00:00:00 GMT

Anton Alexa — We develop a dynamical method for proving the sharp Berezin-Li-Yau inequality. The approach is based on the volume-preserving mean curvature flow and a new monotonicity principle for the Riesz mean R_Λ(Ωₜ). For convex domains we show that R_Λ is monotone non-decreasing along the flow. The key input is a geometric correlation inequality between the boundary spectral density Q_Λ and the mean curvature H, established in all dimensions: in d=2 via a near-disk Fourier analysis, and in d≥ 3 via the boundary Weyl expansion together with a local spectral rigidity argument near the ball, with a first-zero exclusion principle closing the global step. Since the flow converges smoothly to the ball, the monotonicity implies the sharp Berezin-Li-Yau bound for every smooth convex domain. As an application, we obtain a sharp dynamical Cesàro-Pólya inequality for eigenvalue averages.

Linear stability and instability of Kähler Ricci solitons

Wed, 19 Nov 2025 00:00:00 GMT

Keaton Naff, Tristan Ozuch — We show that the recently discovered BCCD shrinking soliton is linearly unstable, by extending the approach of and, via recent work the on gradient shrinking Ricci solitons. On the other hand, we prove that the weighted L²-spectra of the weighted Lichnerowicz Laplacians of steady and expanding Kähler Ricci solitons are nonpositive in real dimension 4. We additionally determine the linear stability of the orbifold singularities of Kähler solitons: shrinkers are unstable, steadies are neutrally stable and expanders are strictly stable. All of these results follow from new Weitzenböck formulae for the weighted Lichnerowicz Laplacian specialized to Kähler metrics.