Papers

We study weighted Carlson-Levin inequalities on cones, computing the optimal constant and classifying optimizers via an Entropy method. The limiting case of a logarithmic Carlson-Levin inequality is also treated. As a Corollary, we obtain a rigidity result for a weighted Gagliardo-Nirenberg inequality. We conclude with some examples involving monomial weights.

We give an explicit description of the scattering matrix for the Hodge-Laplacian on co-closed p-forms on asymptotically conic manifolds of dimension $n\geq 3$. We develop generalized eigenfunctions and a functional calculus to describe the result, a departure from the Fourier Integral Operators used in the scalar case. Starting from the Hodge decomposition at infinity, we construct generalized eigenforms which are co-closed and establish a spectral representation for the p-form Hodge Laplacian. The special case of dimension 3 for $p=1$ characterises the electric field for Maxwell's equations.

We show a priori bounds for the generalized Parabolic Anderson Model $(\partial_t - \Delta)u = \sigma(u) \diamond \xi$ in finite volume in the full subcritical regime. The approach is based on the observation that, for constant initial data, the time interval over which the solution remains close to its initial condition can be chosen uniformly in the value of the initial condition. A comparison principle is then used to extend this estimate to general bounded initial data.

We analyze pattern formation from a homogeneous steady state in a one-component nonlocal adhesion model with population pressure and degenerate mobility. First, using linear stability analysis, we derive the instability threshold and a selection rule for the fastest-growing mode, and elucidate the mechanism by which the selected wavenumber shifts toward lower wavenumbers as the mean density increases. We then perform a weakly nonlinear analysis near the critical adhesion strength and derive an explicit expression for the Landau coefficient in the Stuart–Landau equation. This expression shows that the critical bifurcation is classified as supercritical or subcritical according to the mean density, the nonlinear exponent, and the second-harmonic response ratio of the kernel. Furthermore, we show that a large nonlinear exponent promotes a transition to subcriticality and confirm, through numerical bifurcation analysis and time-dependent simulations, a bifurcation structure with a fold point and the formation of mesa patterns. Finally, through the energy limit as \(m\to\infty\), we relate the observed mesa profiles to a capacity-constrained limiting structure.

Orthonormal Strichartz estimates for the free Schrödinger propagator take the form \begin{equation*} \left\Vert\sum_j\lambda_j\left|e^{it\Delta}f_j\right|^2\right\Vert_{L_t^\frac{q}{2}L_x^\frac{r}{2}(\mathbb{R}\times\mathbb{R}^d)}\lesssim\Vert\lambda\Vert_{\ell^\alpha(\mathbb{C})} \end{equation*}for arbitrary orthonormal systems $(f_j)_j$ in the homogeneous Sobolev space $\dot{H}^s(\mathbb{R}^d)$. In the admissible region, the optimal range of $\alpha$ has been established when $q\geq r/d'$. For $d\geq2$ and $2<q<r/d'$, it has been an open question as to determine whether the estimate holds when $\alpha=q/2$. We prove that the estimate holds in this critical case whenever $q=4$ and $r<\infty$. Our approach is robust and we illustrate this by extending the result to fractional Schrödinger propagators.

We establish a priori interior curvature estimates for the scalar curvature equation in dimension 4. Our approach relies on the doubling method introduced by Shankar and Yuan [SY25]. Key to the proof are an a priori doubling inequality under a small gradient assumption and the Alexandrov regularity for viscosity solutions; the latter is obtained via a new iterative rotation argument. Furthermore, we establish interior regularity for $C^1$ viscosity solutions.

We study the one-dimensional quasilinear Majda–McLaughlin–Tabak (MMT) equation on a large torus $[0,L]$: \begin{align*} i \partial_t u +2\pi|\nabla|^\sigma u +\lambda^{2}|\nabla|^\beta\left[ \left||\nabla|^\beta u\right|^{2} |\nabla|^\beta u\right]=0. \end{align*} Our focus is on the well-posedness of its dynamics and the emergence of kinetic behavior where the domain size $L$ tends to infinity and the nonlinearity $\alpha=\lambda^2L^{-1}$ vanishes. In contrast to semilinear dispersive models, the quasilinear structure leads to unavoidable derivative loss, which prevents the construction of solutions via iteration of the Duhamel formula. Our results exhibit a dichotomy depending on the dispersion exponent $\sigma$. For $\sigma\in(1,2]$, we prove that, with high probability, solutions exist up to time scales $T_0 \sim \alpha^{-\frac54+} \wedge \alpha^{-\frac1{1-\beta}+}$, and that only trivial resonances occur, leading to a degenerate wave kinetic equation. For $\sigma\in(0,1)$, we prove the existence up to time scales $T_0 \sim \alpha^{-1-}$ and show that the second-order statistics are well approximated by the wave kinetic equation. In both cases, the solutions remain smooth while exhibiting smallness in suitable $L^\infty$-based norms despite having large total energy. The proof proceeds in two main steps. First, we establish the propagation of randomness for a suitably truncated equation, which allows us to overcome the derivative loss and recover the kinetic description. Then, we perform deterministic high-order energy estimates and a bootstrap argument to extend the solution up to time $T_0$.

We study the quantitative fast and slow weak flocking of the phase-spatially extended kinetic Cucker-Smale model in confining potential fields. The confining potential is allowed to be nonconvex. In contrast to the phase-spatially confined case, the communication weight may have no positive lower bounds, while spatial and velocity diameters may remain infinite. Moreover, an indefinite Hessian of confining potential prevents the direct use of convexity-based coercive estimates. To overcome these difficulties, we identify three structural assumptions that are sufficient for the noncompact hypocoercive method: quadratic confinement, a globally Lipschitz force, and a strict virial inequality. The admissible class of confining potentials includes genuinely nonconvex radially symmetric ones and localized oscillatory perturbations of the harmonic potential. Weak flocking analysis combines three key ingredients: a microscopic mechanical-energy estimate, a time-varying effective region, and a macroscopic Lyapunov functional. For this, we first establish global existence for initial data with finite second moments. In the exponential mechanical-energy class, we further show the uniqueness and finite-time Osgood-type stability in 1-Wasserstein distance. For polynomially decaying initial mechanical-energy tails, we derive the optimal algebraic decay exponent of the fluctuation energy throughout the admissible communication-decay regime. The optimality is verified by a symmetric countably infinite particle solution for a fixed nonconvex potential. For exponentially decaying tails, a two-stage localization argument yields exponential weak flocking on an optimal exponential time scale. These results show that well-posedness and weak flocking persist even for fully noncompact data under genuinely nonconvex confining forces.

Motivated by Landis's conjecture on unique continuation at infinity for the Laplacian, we study the corresponding property for the Cauchy-Riemann operator. We prove that every weak solution of $\bar\partial u=Vu$ on a neighborhood of infinity, with $V\in L^\infty$, vanishes identically if it decays exponentially at a rate greater than $ 2\|V\|_{L^\infty}$. This conclusion is sharp both in the constant $2\|V\|_{L^\infty}$ and in the order of exponential decay required. More generally, we establish unique continuation at infinity for a broad class of radially decaying bounded potentials, with optimal decay rates determined by the decay of the potential. We also obtain related unique continuation results for $L^2$ potentials and for compactly supported potentials under weaker assumptions at infinity.

We establish Harnack-type inequalities for local minimizers of nonlocal double phase functionals, involving an explicit two-radius positive-part tail and a negative far-field tail. The key difficulty is the mismatch between the natural \((p,q)\)-growth scales in the available modular supremum estimate and the low integrability exponent provided by the weak Harnack inequality. To overcome this obstacle, we establish a concentric modular estimate and derive from it a local upper estimate valid for every integrability exponent. To the best of our knowledge, our results provide the first Harnack-type theory for such nonlocal double phase functionals under the natural structural conditions.

Spatially restricted immune access can undermine tumour control even when total immune recruitment appears sufficient. We develop a nondimensional reaction–diffusion–chemotaxis model for hepatocellular carcinoma that couples tumour growth and immune-mediated killing with effector diffusion, saturating chemokine-dependent recruitment, and migration along an effective CXCL9/CXCL10/CXCL11–CXCR3 signal. We establish nonnegativity, uniform tumour and mass bounds, and global boundedness of classical solutions in one dimension and, in arbitrary dimensions, when chemokine production is tumour-driven. Analysis of the homogeneous dynamics shows that the threshold $\sigma_0>\delta$ is only local and does not preclude bistable tumour persistence. A mode-wise dispersion relation identifies stationary finite-wavelength and oscillatory instabilities, whose critical sensitivities, dominant modes, growth rates, and frequencies are quantitatively recovered by conservative finite-volume simulations. Using matched initial states and identical mean recruitment, we show that a well-mixed model may predict clearance while margin-limited recruitment preserves a stable interior tumour refuge. Chemotaxis improves interior effector access and reduces, but need not eliminate, this refuge. Spatial-overlap biomarkers further connect model-generated phenotypes to pathology and spatial-omics observables. These results distinguish immune abundance from effective spatial access and identify access-limited recruitment as a mechanism of incomplete tumour control.

This note makes available, in facsimile form, a set of handwritten notes from September 1993 devoted to Bochner-type maximum principle computations for a general class of quasilinear parabolic equations which contains, as a special instance, the normalized parabolic $p$-Laplacian \[ u_t = \Delta u + (p-2)\,|\nabla u|^{-2}\,\nabla^2u(\nabla u,\nabla u). \] The notes were acknowledged in the introduction of the paper of Jin and Silvestre \cite{JS}, and the scheme they propose has recently been implemented in \cite{BGM} to obtain the sharp Li-Yau inequality for this equation. They are reproduced here verbatim, with no alterations, preceded by a brief introduction explaining their content and the subsequent history of the equation.

We study the spatiotemporal dynamics of the expansion of a bacterial colony on a hard substrate. Nutrient from the substrate diffuses into the colony and is taken up by the bacterial cells for them to grow and divide, expanding the initial monolayer and then pancake-shaped colony. The concentration of nutrient determines the local cell growth rate in the colony. Mass conservation relates such local growth rate with velocity which is approximated to be proportional to the pressure gradient by Darcy's law. Altogether, the growing colony is modeled as a moving-boundary problem with the nutrient concentration and pressure solving a reaction-diffusion equation and Laplace's equation, respectively. We analyze the self-consistent moving-boundary model with respect to different geometrical setting. For a general three-dimensional cylindrically symmetric model, we study the steady-state nutrient concentration. We construct and analyze a one-dimensional model for vertical expansion and a two-dimensional disk model for radial expansion of the colony. Our analysis finds that the nutrient depletes into the colony and slows down the vertical expansion of the colony. The vertical level where the nutrient concentration reaches the Monod constant, a threshold below which individual bacteria hardly grow, lowers down exponentially fast. We also estimate the asymptotic radial expansion rate. Moreover, we establish that the region where the nutrient concentration is above the threshold, allowing bacteria to grow and the colony to expand radially, is a ring-shaped peripheral region of fixed thickness. All these are consistent with experiment and agent-based simulations reported in literature.

We introduce a version of the Monge–Ampère equation on finite graphs, motivated by nonlinear graph-based interpolation and semi-supervised learning. The operator is defined as the product of discrete analogs of the Hessian eigenvalues, obtained via local order statistics of function values at neighboring vertices. We derive an equivalent Bellman-type formulation of the inhomogeneous Dirichlet problem, establish a comparison principle and uniqueness in the strictly graph-convex class, and investigate existence via Perron's method, identifying certain graph-theoretic obstructions. We also study the homogeneous equation, for which the problem reduces to a nonlinear interpolation rule involving the smallest discrete eigenvalue. Finally, we propose numerical schemes for both the homogeneous and inhomogeneous problems.

We study the spatially homogeneous Boltzmann equation without angular cutoff for very soft potentials satisfying the inverse power law relation $\gamma+4s=1$. Our main results establish the existence, uniqueness, stability and regularization estimates of solutions in the critical Sobolev space $ H^{-(\gamma + 2s + \frac{3}{2})} $ with a logarithmic correction. Combined with the recently established monotonicity of the Fisher information, the solutions extend globally in time. Our primary tools are energy estimates based on a simultaneous dyadic localization in the phase and frequency variables, together with sharp commutator estimates between the collision operator and the localization operators.

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 $2\leq n \leq 4000$ is verified by a rigorous computer-assisted argument using interval arithmetic.

We study the existence and classification of local $(-1)$-homogeneous axisymmetric solutions $(u, p)$ of the three-dimensional incompressible stationary Navier-Stokes equations near a singular ray. We focus on such solutions with Type II behavior, which satisfy $0<\limsup_{x\in\bS^2, x\to P}|u|/|\ln \text{dist}(x, P)|<\infty$, where $P$ is the south or north pole. Apart from the Landau solutions, these are the least singular nontrivial $(-1)$-homogeneous axisymmetric solutions. We construct a four-parameter family of local $(-1)$-homogeneous axisymmetric solutions with at most logarithmic growth near the singular ray and represent them as convergent series whose coefficients are determined recursively. Conversely, we prove that every local $(-1)$-homogeneous axisymmetric solution satisfying $|u|=o(\text{dist} (x, P)^{-1})$ as $x\to P$ on $\bS^2$ belongs to this family. In particular, this family contains Type II solutions with nonzero swirl, in contrast to the global $(-1)$-homogeneous axisymmetric Type II solutions in $\R^3\setminus\{x'=0\}$, which have no swirl. We also establish derivative estimates for the constructed solutions and identify the singular force generated by these solutions across the singular ray in the sense of distributions.

The observable of the Talbot effect is the intensity behind the grating: the density of the evolving field, not the field itself. Its fractality was previously known only for step data with rational jumps. We prove the general case. For arbitrary real data of bounded variation with at least one jump, the density of the free Schrödinger evolution on the torus is fractal at almost every time, with graph of upper box dimension exactly three halves. Its critical Sobolev mass diverges logarithmically, at a rate given in closed form by Wiener's jump statistic of the datum and independent of the positions and phases of the jumps. The same rate is halved along time traces and doubled for the Airy flow, so the critical mass probes the dispersion relation. Transported by the rank calculus of derangetropy operators, the theory equips every probability law with a quantum carpet: Gauss-sum revivals on quantile cells at rational times, a universal fractal at almost every other. The spectral growth rate of a single measured intensity profile returns the jump content of the grating, a prediction open to test in optical and matter-wave interferometry.

Let $n\ge2$ and let $\Omega\subset\mathbb R^n$ be an arbitrary bounded open convex set. The author prove that, for $p>n$, the Dirichlet problem \[ \det D^2u=(-u)^p\quad\text{in }\Omega, \qquad u=0\quad\text{on }\partial\Omega, \qquad u>0\quad\text{in }\Omega \] has at most one convex Alexandrov solution. The proof is based on the affine behavior of the Monge–Ampère energy and on a power-concavity property of the $L^{p+1}$ mass along the Legendre path connecting two solutions. At the homogeneous exponent $p=n$, the same argument shows that any two nonzero solutions with the same coefficient are positive multiples of one another.

We prove the rigidity conjecture of [16, Conjecture 1.2] for smooth strictly convex capillary Gauss solitons in a Euclidean half-space with an acute contact angle: every such soliton is a spherical cap. Combined with our previous convergence result for the capillary Gauss curvature flow [16, Theorem 1.1], it follows that the flow starting from a strictly convex capillary hypersurface with an acute contact angle converges to a capillary spherical cap, after a suitable rescaling.

Let $(v,p)$ be a smooth stationary Navier–Stokes solution in $\mathbb R^3$ with $v(x)\to0$ as $|x|\to\infty$. We prove that $v\equiv0$ if its vorticity $\omega:=\nabla \times v$ belongs to $L^{s,\infty}(\mathbb R^3)$ for some $9/5\le s\le1.87$. This gives a global weak-Lorentz vorticity criterion beyond $9/5$ requiring neither smallness nor an additional Fubini-type hypothesis. The finite Dirichlet integral is recovered from the vorticity hypothesis rather than assumed. In particular, $|\omega(x)|=O(|x|^{-\alpha})$ implies triviality for $\alpha\ge300/187$, crossing the $|x|^{-5/3}$ vorticity-decay scale. The proof combines this automatic finite-energy upgrade with a new variable-denominator Bernoulli–vorticity quotient identity. Its completed-square form gives nonnegative defect terms, and localizing the concavity of the denominator creates a positive measure on a Bernoulli level surface. A pressure-weighted convex extension of this identity yields a quantitative normalized-vorticity trace and quotient-gradient control. These estimates are then coupled to the Bernoulli-gradient identity through a nonlinear feedback estimate.

We consider the $1$-equivariant harmonic map heat flow (HMHF) from $\mathbb R^2$ to $\mathbb S^2$ and the radial energy-critical nonlinear heat equation (NLH) in dimensions $d=3,4,5$. We prove that every finite-energy solution of (HMHF) and every $\dot H^1$-bounded solution of (NLH) has at most one bubble: every finite-time blow-up has exactly one bubble, whereas every global solution has either no bubble or one bubble at infinite time. Starting from the soliton resolution, we exclude bubble trees by modulation analysis, deriving a contradiction from the relative dynamics of the two innermost scales. This energy method does not rely on maximum principle and, in particular, applies without restriction on the bubble signs.

In this paper, we show how to adapt a derivation of Hartree equations due to Knowles and Pickl in the case where the one-particle hamiltonian is an Anderson-Hermite operator in dimension 1 and 2 or a formal Anderson-Hermite operator in dimension 3. By proving energy estimates for solution of the limit equation, uniform in the noise in bounded sets, we achieve a double limit. Namely, we show that we can commute the mean-field limit with a limit in noise.

In this paper we establish a criterion on weak type $(1,\,1)$ bound of the following rough singular integral $$T_{\Omega,K,a}f(x)={\rm p.v.}\int_{\mathbb{R}^n}\Omega(x-y)K(x,y)m_{x,y}a f(y)dy,$$ where $m_{x,y}a=\int_0^1a(sx+(1-s)y)ds$ with $a\in L^1(\mathbb{R}^n)$ and $\hat{a}\in L^1(\mathbb{R}^n)$, $\Omega$ is homogeneous of degree zero, integrable in $\mathbb{S}^{n-1}$ and satisfies the cancellation condition $\int_{\mathbb{S}^{n-1}}\Omega(\theta)d\sigma(\theta)=0$ and $K$ is a measurable function defined on $\mathbb{R}^n\times\mathbb{R}^n\setminus \{(x,x):x\in\mathbb{R}^n\}$ and satisfies a Hölder condition. By assuming that $\Omega\in L\log L(\mathbb{S}^{n-1})$ and the operator $T_{\Omega, K}f(x)={\rm p.v.}\int_{\mathbb{R}^n}\Omega(x-y)K(x,y)f(y)dy$ is bounded on $L^2(\mathbb{R}^n)$, we prove the weak type (1,1) bound of $T_{\Omega,K,a}$. As several applications, we obtain a large class of singular integral operators which possess weak type $(1,\,1)$ bound. The main results of this paper essentially extend and generalize some known ones.

In this paper, we investigate the asymptotic behavior of a coupled fluid–rigid body system, consisting of a rigid body immersed in a three-dimensional viscous incompressible fluid, as the body size and the fluid viscosity vanish. We prove that the limiting velocity field is governed by the whole-space incompressible Euler equations. Moreover, the influence of the fluid on the rigid body motion is shown to vanish in this limit.

In the unit ball, we study a semilinear system that gives rise to a free boundary of Alt-Phillips type, where the equation is governed by a mixed variable-exponent operator of $(p,q)$-Laplacian type. Under certain structural conditions, we establish the existence of nonnegative radial solutions that are $C^1$, with their norms depending on the structural data.

Let $\Omega\subset\mathbb R^3$ be a smooth bounded domain and let $g:\partial\Omega\to\mathbb S^2$ be smooth. We prove that any global minimizer of the Oseen–Frank energy subject to the strong anchoring condition $n=g$ is smooth in a full neighborhood of $\partial\Omega$. The result holds for arbitrary positive splay, twist, and bend constants, without any small-anisotropy assumption. It resolves the boundary-regularity part of Lin and Liu's Problem~(c) for the pure Oseen–Frank problem and for its prescribed smooth magnetic-field perturbation.

We establish sharp Barron regularity for Coulombic many-electron wave functions after extraction of the universal cut-off Jastrow factors. Following the factorization of Fournais et al.~\cite[Definition~1.4]{FournaisEtAl2005}, for a Coulombic eigenfunction $\psi$ we define the successive quotients by \[ \phi=e^{-F_{2,\mathrm{cut}}}\psi \quad\text{and}\quad \phi_3=e^{-F_{3,\mathrm{cut}}}\phi=e^{-(F_{2,\mathrm{cut}}+F_{3,\mathrm{cut}})}\psi. \] Then \[ \phi,\phi_3\in\mathcal{B}^s(\mathbb{R}^{3N}) \qquad\text{for every }s<2. \] This range is optimal among universal factorizations. No factor depending only on the particle number and the nuclear data, but not on the eigenfunction or its eigenvalue, can make every corresponding quotient belong to $\mathcal{B}^2$. We also determine the exact endpoint growth. Writing $\varepsilon=2-s$, we prove that, for either $u=\phi$ or $u=\phi_3$, there is a computable constant $M$ independent of $\varepsilon$ such that \[ \left\|u\right\|_{\mathcal{B}^{2-\varepsilon}}\leq\frac{M}{\varepsilon^2}\left\|u\right\|_{\mathcal{B}^1}. \] For the unperturbed two-electron atom we prove, with a constant independent of $\varepsilon$, \[ \left|\left\|\phi_3\right\|_{\mathcal{B}^{2-\varepsilon}}-\frac{32\pi Z\lvert\phi_3(0,0)\rvert}{\varepsilon^2}\right|\leq\frac{C}{\varepsilon}. \] Hence the quadratic rate in the upper bound is sharp whenever $\lvert\phi_3(0,0)\rvert\neq0$, as is the case for the ground state.

We develop a componentwise geometric and monodromy theory for the one-support generalized Lamé equation on an elliptic curve, with singularities at \(0\) and \(\pm p\). Its log-free curve decomposes canonically into irreducible even and non-even components, the former being governed by elliptic Painlevé~VI. For the non-even component, we construct the hyperelliptic spectral curve, Baker–Akhiezer functions, and addition map, and prove that \[ \deg \sigma_{n,p}^{(1)}=n(n+1). \] After quotienting by the involution \(T\mapsto -T\), we identify the non-even spectral curve with the classical Lamé spectral curve of weight \(n\), compatibly with the addition map and the rational function \(\kappa\). This identification is realized by \[ \widetilde B=T^2-n(n+1)\wp(p), \] and associates every non-even generalized Lamé equation with a unique classical Lamé equation on the same elliptic curve having equivalent period monodromy. Fixing \(\widetilde B\) yields an isomonodromic deformation with \(\tau\) fixed. Together with the Painlevé-VI deformation on the even component, it gives a componentwise interpretation of the collision \(p\to0\), and yields a finite descent on the admissible completely reducible locus. The classical spectral, finite-gap, finite-monodromy, and curvature theories consequently transfer to the non-even component. Finally, within the symmetric family $\left(n_0,n_1,n_2,n_3,\frac12,\frac12\right),$ the one-support case forms an affine genus-zero hierarchy, whereas for $\left(1,1,0,0,\frac12,\frac12\right)$, the non-even normalization is generically elliptic and becomes rational on the discriminant locus, while the full compactified log-free curve retains arithmetic genus two. This first genus jump marks the boundary of the affine theory and motivates a genus-dependent componentwise geometry.

This paper is concerned with the asymptotic behavior of the principal eigenvalue $\lambda(D)$ of the elliptic eigenvalue problem \[ -D\Delta_{M}u - a\langle \nabla_M f, \nabla_M u\rangle_g + c u = \lambda(D)u, \] posed on a closed orientable Riemannian manifold $(M,g)$, in the small-diffusion limit $D \to 0^+$. Under the assumption that $f$ is a Morse function on $M$, we establish that the limiting value $\lim_{D\to 0}\lambda(D)$ is completely characterized by the critical points of $f$ and the associated Riemannian Hessian, specifically through the values of $c$ and the Riemannian Hessian eigenvalues at those points.