Papers

This paper is devoted to a new two-dimensional nonlinear dispersive wave model named as the Camassa-Holm-Zakharov-Kuznetsov (CH-ZK) equation which combines the nonlinear structure of the Camassa-Holm equation with the transverse Laplacian dispersion of the Zakharov-Kuznetsov equation. We first establish the local well-posedness of its Cauchy problem in a suitable Sobolev space and derive a blow-up criterion for strong solutions. Then the finite-time blow-up strong solutions for the CH-ZK equation have been constructed. Moreover, we prove a unique continuation property for the solutions to the CH-ZK equation. Finally, we investigate the existence of both peaked and smooth solitary waves, and obtain a rigidity theorem for the traveling wave solutions according to the magnitude of wave speed.

We study large mass monopoles with structure group $\mathrm{SU}(2)$ or $\mathrm{SO}(3)$ on asymptotically conical $\mathrm{G}_2$-manifolds and Calabi–Yau $3$-folds, with fixed asymptotic class. After placing the AC asymptotic theory, the variational compactness theory of Parise–Pigati–Stern, and Li's singular abelian compactness theory in a common $\Theta$-monopole framework, we prove that the mass-renormalized Yang–Mills–Higgs and intermediate energy measures converge to $8\pi\|T\|$ for a compactly supported calibrated integral codimension-three cycle $T$. This identifies the two limiting currents and shows that the variational calibration inequalities are saturated. Using this common limit as the starting point for a finer analysis, if $\mathcal S$ is the calibrated support, $\mathcal Z$ the Kuratowski upper limit of the Higgs zero sets, and $\mathcal C$ the limiting nonabelian locus, defined as the Kuratowski upper limit of Li's curvature concentration loci, then $\mathcal S\subset\mathcal Z\subset\mathcal C=\mathcal S\cup\mathcal O$, where $\mathcal O$ is precisely the obstruction to effective codimension-three monotonicity. For the cohomogeneity-one large mass families on the Bryant–Salamon $\mathrm{G}_2$-manifolds and the Stenzel Calabi–Yau $3$-fold, we prove that $\mathcal O=\varnothing$. On $X\setminus\mathcal C$ the sequence abelianizes; corrected longitudinal curvatures converge smoothly, and the remaining compactness alternatives are governed by $L^2$-harmonic $2$-forms.

The Schiffer conjecture states that if a smooth domain $\Omega \subset \mathbb{R}^n$ admits a Neumann eigenfunction of the Laplacian which is constant at the boundary, then the domain is a ball. It is intimately related to Pompeiu's problem, stating that if a nonzero function integrates zero over any rigid motion of $\Omega$, then $\Omega$ is a ball. We disprove both conjectures in $\mathbb{R}^2$, constructing infinitely many planar domains $\Omega$ which are not balls and satisfy the conditions above. Our domains are $N$-fold symmetric, with $N$ sufficiently large. Our approach is based on a novel strategy of considering a relaxed problem where $N$ can be any real number (which corresponds to the Schiffer problem only when $N$ is a natural number). We then apply bifurcation theory to this relaxed problem, showing that the size of the local bifurcation branch can be taken independently of $N$. This result allows us to conclude that branches starting with $N$ sufficiently close to an integer reach integer values of $N$.

We introduce a purely operator-theoretic framework for quantitative homogenization that bypasses the traditional reliance on large-scale spatial regularity and probabilistic assumptions. Inspired by Tartar's vision of a point-free theory, we derive explicit norm resolvent estimates using only the algebraic structure of multiscale operators and the abstract geometry of Hilbert spaces. In this framework, the effective macroscopic dynamics and the abstract corrector emerge naturally from an orthogonal decomposition of the state space, governed algebraically by a Schur complement. To quantify the convergence rate, we introduce a frequency-splitting technique and solve a generalized Sylvester equation that controls the commutator between the differential structure and the highly oscillatory material properties. This abstract perspective unifies stationary, non-stationary, periodic, quasi-periodic, and stochastic homogenization. We demonstrate that the physical distinctions between these media–and their respective convergence rates–are entirely captured by the behavior of the spectral measures of the microscopic and macroscopic derivative operators near zero frequency.

We study the law of the exit time and exit point of a Brownian motion in a two-dimensional domain with reflecting boundary conditions, except on small disjoint exit windows through which the stochastic process can escape the domain. In the limit of infinitely small exit windows, it is natural to assume that the process starts from the quasi-stationary distribution. In this setting, we obtain a precise description of the exit event.

We prove that the standard degree-two and degree-three vortex solutions of the Ginzburg-Landau equation are nondegenerate. Their Morse indices are also computed. The proof relies on new explicit upper and lower bounds of the modulus of these solutions and a comparison argument. It is expected that our method can be generalized to study higher degree solutions.

We study an inverse problem of recovering a time-dependent convection term and density coefficient of the convection-diffusion equation from partial data on a certain type of compact Riemannian manifold of dimension at least three. We prove that the knowledge of a partial input-output operator determines both coefficients uniquely up to a natural gauge. Our geometric setting is conformally transversally anisotropic manifolds, that is, compact Riemannian manifolds with boundary that are conformally embedded into a product of the Euclidean line and a transversal manifold. Additionally, we assume that the attenuated geodesic ray transforms of one-forms and functions are both injective on the transversal manifold.

We determine the complete local Bloch spectrum bifurcating from the origin for small-amplitude periodic hydroelastic Stokes waves in infinite depth, under the combined effects of gravity, surface tension, and elastic bending. Away from the Wilton-type resonance set, we construct a real-analytic Stokes-wave branch and analyze the four eigenvalues emerging from the defective zero eigenvalue of the linearized hydroelastic Euler system. Using analytic spectral perturbation theory and Hamiltonian-reversible reductions, we decouple them into a Benjamin–Feir pair and a long-wave pair. The long-wave pair remains purely imaginary and has the singular scale $\cO(\sqrt{|\mu|})$, whereas the Benjamin–Feir pair is governed by an explicit discriminant whose leading sign yields a sharp criterion for modulational stability and instability. We derive the exact non-resonant phase diagram in the surface-tension-bending parameter plane and identify a bounded stability island generated by elastic bending. In the unstable region, and away from a drift degeneracy, the Benjamin–Feir branches form a local figure-eight curve. In the zero-bending limit, the reduced coefficients recover the known deep-water gravity and gravity-capillary results, while the change from the finite-depth $\cO(|\mu|)$ long-wave scale to $\cO(\sqrt{|\mu|})$ shows that the infinite-depth problem is singular.

We study local minimizers of a sextic-potential Landau–de Gennes energy for nematic liquid crystals in the small-elastic-constant limit. Under the uniform energy and $L^\infty$ bounds, these minimizers converge to a locally energy-minimizing harmonic map $\mathbf{Q}_0$ into a biaxial vacuum manifold. The main result of this paper is that such a limiting map $\mathbf{Q}_0$ has no interior point singularities. The proof relies on a geometric identification of the lifted Frobenius metric on the universal cover $\mathbb{S}^3$ with a rescaled Berger metric. For a hypothetical tangent cone at a point singularity, its link is a nonconstant harmonic two-sphere into the Berger sphere. We construct a smooth variation field adapted to the Hopf direction and show that it induces a quantitative instability estimate, in contradiction with the shifted stability inequality inherited from local minimality. This replaces the usual round-sphere test fields by a target-specific construction and rules out interior point defects in the biaxial setting. The result is sharp in view of the well-known existence of point defects in the uniaxial theory.

Extreme flows realize the largest possible growth, either instantaneously or in finite time, of certain quantities of interest which is achieved by a suitable choice of the initial condition or the applied forcing. The quantities of interest usually measure some small-scale properties and therefore provide information about the regularity of the flow. Extreme behavior is at the heart of several open problems in fluid mechanics including the dissipation anomaly in turbulence and formation of singularities in various models of fluid flow. In this essay we describe a framework making it possible to study such extreme behavior systematically by combining mathematical analysis, scientific computation and physics. As a first step, one aims to deduce rigorous upper bounds on the growth of the quantities of interest in the solutions of a given model. These inequalities express fundamental limitations on the most extreme behavior possible among all admissible solutions. However, given how they are obtained, these bounds may be conservative and overestimate the growth actually realizable in the system. In order to probe this possibility, as the next step, we set up variational optimization problems where the growth of the quantity of interest is maximized under suitable constraints. Solution of such problems is enabled by modern methods of numerical optimization. When properties of the thus obtained maximizers match the bounds, the bounds are declared sharp and therefore cannot be fundamentally improved. Finally, properties of the solutions saturating the bounds reveal insights about the physical mechanisms realizing the extreme behavior. We survey problems where this research program has produced sharp bounds together with extreme flows saturating these bounds. A collection of open problems is then presented and we close the essay with a discussion of possible methodological improvements.

Grayson and Gage proved that an immortal curve shortening flow of simple closed curves on a closed surface converges subsequentially to a closed geodesic, and they asked whether this limiting geodesic is unique. We answer this question negatively by constructing a smooth Riemannian metric on $\mathbb{S}^2$ and an immortal simple closed curve shortening flow that converges along different sequences of times to every geodesic in a one-parameter family of distinct simple closed geodesics.

We study a parabolic-parabolic chemotaxis system motivated by microglial recruitment toward an amyloid-$\beta$-associated signal in Alzheimer's disease. The signal is subject to a nonnegative Radon measure-valued Neumann influx on a vascular portion of the boundary, while microglial cells respond to a nonlocal spatial average of the signal. For a fixed sensing length $\sigma>0$, the chemotactic velocity is $b_\sigma[v]=\nabla K_\sigma[v]$. For every fixed $\sigma>0$, the nonlocal operator maps finite signal mass into a bounded spatially Lipschitz velocity field. We introduce a weak-solution concept adapted to the low regularity induced by the boundary measure and construct a fully implicit upwind finite-volume approximation in which the boundary source is discretized through its exact mass on each boundary face-time cell. We establish existence and positivity of discrete solutions, together with uniform mass, energy, discrete-gradient, and compactness estimates. Finally, we prove subsequential convergence of the discrete solutions toward a nonnegative weak solution of the continuous problem.

Consider the inverse problem of recovering a strictly convex conical obstacle in $\mathbb{R}^3$ from the diffraction coefficients along with arrival directions (lens data) or arrival times of diffracted waves. The incident wave is a spherical pulse emanating from a point, and the measurements of diffracted waves are taken at an arbitrarily sized receiver placed within the reflection shadow. Specifically, the lens data or arrival times determine the location of the tip, whereas the diffraction coefficients reconstruct the shape of the cone. Since diffraction coefficients are described by half waves over the complement of the cone base in $\mathbb{S}^2$, we reduce inverse diffraction by a cone in $\mathbb{R}^3$ to identifying the reflected wavefront in $\mathbb{S}^2$ and recovering the obstacle using reflected rays on the sphere. The former is accomplished by constructing the Hadamard parametrix for half waves near the wavefront, whereas the latter relies on the topological properties of broken geodesics on $\mathbb{S}^2$. The framework developed in this paper exploits the analytic and geometric structures of diffracted wave fields characterized in the Geometrical Theory of Diffraction, and establishes, for the first time, a rigorous inverse theory corresponding to GTD.

In this paper, we are interested in considering the semi-linear damped wave equation with the power nonlinearity of derivative type in $2$D exterior domains to indicate the global (in time) existence of small data solutions, which has never appeared in previous studies. Our proof is based on constructing a judicious time-weighted function space and demonstrating nonlinear estimates compatible with fractional powers of the Dirichlet Laplacian linked to the semigroup decay structure. Moreover, the large time behavior of the time derivative of the obtained global solutions and its sharp estimate are also discussed in this paper.

We study the Cauchy problem for the nonlinear Schrödinger equation on $\mathbb T^d$ with random initial data and a general non-algebraic power-type nonlinearity. We establish almost sure local well-posedness in every spatial dimension and for the whole mass-supercritical range allowed by the natural condition $0<s_{\mathrm c}<1+a$. The main new ingredient is a frequency-gaining probabilistic refinement of the Galilean bilinear estimates recently developed by Kwak and Kwon \cite{KwakKwon}. In the random setting, the gauge decomposition gives rise to three new types of terms: a mean-free coefficient, an opposite-phase interaction, and a scalar remainder. We control them by new resonance counting and large deviation arguments, and close the local theory through a phase-adapted two-component contraction. In the energy-critical case, our result extends the low-dimensional algebraic theories of Nahmod–Staffilani \cite{NahmodStaffilani15} and Yue \cite{Yue21} to every dimension $d\geq3$, including the higher-dimensional non-algebraic models.

In this paper, we investigate a stochastic two species competition Keller–Segel model. Chemotactic movement, interspecific competition, and environmental fluctuations may act simultaneously when two species respond to the same chemical signal. We consider a stochastic two-species chemotaxis–competition system on a bounded smooth domain $\mathcal O\subset\mathbb R^n$, $n\in\{1,2\}$. Both species contribute to the production of the signal and move along its gradient; their population equations also include nonlinear competition and multiplicative noise. If the self-damping exponents exceed two, we prove that the system has a unique global adapted nonnegative mild solution. We also establish continuous dependence in probability on admissible initial data over finite time intervals. The results prove the global well-posedness under the stated assumptions and we still show that small changes in the initial populations, in probability, to small changes in the solution. Due to the two coupled chemotactic fluxes and the stochastic terms, the $L^p$ estimates contain terms with no fixed sign. They are controlled by parabolic estimates for the signal equation, superquadratic self-damping, and stochastic convolution estimates.

We study the initial-boundary value problem for the Navier-Stokes equations in the half-space with initial data belonging to a suitable weighted Lebesgue space. More precisely, we consider a weighted Lebesgue space associated with a weight function defined as a product of powers of the distances from finitely many fixed points. This framework was introduced in a previous work by A. P. Di Feola and V. Pane (J. Math. Anal. Appl. 558 (2026), 130390) for the study of the initial-boundary value problem for the Stokes system. The present paper completes that analysis and, at the same time, generalizes to the initial-boundary value problem the results obtained by Maremonti and Pane (J. Math. Fluid Mech. 27 (2025), Art. 2) for the Navier-Stokes Cauchy problem. We prove the existence (local) and uniqueness of a smooth solution and derive $L^q$-estimates, with $q>n$, together with the spatial asymptotic behavior of the velocity field.

We establish a sharp well-posedness and norm inflation theory for the Camassa–Holm equation in critical Triebel–Lizorkin $F^{1+1/p}_{p,q}(\mathbb{T})$. At the endpoint $p=1$, we prove local Hadamard well-posedness for $1\le q<\infty$. In contrast, we prove norm inflation for $1<p<\infty$ and $1\le q\le\infty$. We also complement the local well-posedness in the critical Besov spaces and higher-regularity Triebel–Lizorkin spaces. The positive results rely on a Lipschitz stability theorem for the periodic Green operator under degree-one Lagrangian flows. The negative result is based on a nested smooth atomic construction on the torus, adapted from its real-line counterpart.

In this paper, we investigate a stochastic Keller–Segel system with porous medium diffusion and nonlinear chemotactic sensitivity on a bounded one-dimensional domain. The model describes cell aggregation in complex environments, where the dispersal of cells is governed by density-dependent diffusion $\Delta u^{[m]}$, reflecting the combined effects of porous media and population crowding, and the perception of chemoattractants follows Stevens' power law, leading to the nonlinear chemotactic sensitivity $\nabla\cdot(u\nabla v^{[a]})$. In addition, random environmental fluctuations are incorporated through multiplicative noise $u\,dW(t)$, which represents stochastic perturbations in population dynamics. For $a\geq1$ and $m\geq2a+1$, we establish the global existence of martingale solutions, uniform a priori estimates, and preservation of non-negativity. The condition $m\geq2a+1$ reveals a balance between nonlinear chemotactic aggregation and porous-medium diffusion: stronger sensing response requires stronger diffusion to prevent excessive aggregation. The proof combines a decoupled auxiliary system, energy estimates, and a stochastic Schauder–Tychonoff fixed point argument to overcome the difficulties caused by degenerate diffusion, nonlinear drift, and stochastic perturbations.

We investigate the emergence of traveling front solutions in a spatial SIRS epidemic model with diffusion acting on the infected population. The model exhibits a natural slow-fast structure due to the presence of a small parameter governing the loss of immunity, which induces a separation of scales in the dynamics. Using a traveling wave reduction, the PDE system is transformed into a singularly perturbed system of ODEs, which we analyze within the framework of Geometric Singular Perturbation Theory. In the singular limits, we study the fast excursions governed by the layer problem, and the slow evolution close to the critical manifold. In particular, we identify an entry-exit mechanism tracking the transitions between slow and fast regimes, and derive a quantitative characterization of the entry-exit dynamics. Numerical simulations of the full system confirm the validity of the proposed geometric picture. The traveling front is shown to consist of a concatenation of local, fast, and slow segments, in agreement with the theoretical analysis.

Let $\mathcal{H}_d(M)$ denote the space of harmonic functions with polynomial growth of degree at most $d$ on a complete Riemannian manifold $(M,g)$. Yau raised two fundamental questions regarding $\mathcal{H}_d(M)$ on complete manifolds with nonnegative Ricci curvature. The first question is the finite dimensionality of $\mathcal{H}_d(M)$, which was confirmed by Colding and Minicozzi. The second question asks whether a sharp upper bound given by its Euclidean analog $\operatorname{dim}\mathcal{H}_{d}(\mathbb{R}^n)$ holds. We verify that the second question is true on locally conformally flat manifolds. Indeed, one can precisely determine the value of $\dim \mathcal{H}_d(M)$ case by case.

In this paper, we study entire solutions and periodic correctors for the coupled-drift Monge-Ampère equation \[ \det D^2u = \exp\{-a\cdot Du+b\cdot x+V(x)-c_0\}, \quad D^2u>0. \] For $V\equiv0$, we obtain a sharp classification of the whole-space solvability regimes: all entire smooth strictly convex solutions are quadratic when $a=b=0$; no such solution exists when $a\neq0$ and $a\cdot b\le0$; and non-quadratic entire solutions exist when $a=0$ and $b\neq0$, or when $a\cdot b>0$. For the null case $a\neq0$, $a\cdot b=0$, we give a scalar maximum-principle argument in every dimension $n\ge2$. For periodic $V$, we prove existence and uniqueness of the normalized pair $(\psi_A,c_A)$ solving the drifted cell problem \[ \det(A+D^2\psi) = \exp\{-a\cdot D\psi+V-c_A\}, \quad A+D^2\psi>0 \quad\text{on }\mathbb T^n. \] We also prove that any asymptotically quadratic entire solution must satisfy $b=Aa$. If its remainder is bounded, then the solution is the corresponding quadratic-periodic corrector up to an additive constant.

We consider the semilinear heat equation in the unit ball with the exponential nonlinearity and an inhomogeneous term $f$. When $f=0$, it is known that the bifurcation structure of the stationary problem undergoes a qualitative change at the critical dimension $N=10$. This change affects the large-time behavior of solutions to the heat equation, and in particular, the grow-up phenomenon occurs for $N\ge 10$. In this paper, we show that once $f$ exceeds a threshold, the bifurcation structure changes to a type that does not appear in the case $f=0$. The change in the bifurcation structure leads to the disappearance of the grow-up phenomenon beyond the threshold. Moreover, we provide a quantitative characterization of this transition by determining the sharp grow-up rates for $N\ge 11$. In particular, we identify a new dimension-specific phenomenon in the threshold case: a log-log type correction term emerges in the grow-up rate only for $N=11$.

In this paper, we establish optimal Liouville theorems and classification results for the \(k\)-Hessian Lane–Emden equation \[ \sigma_k(-D^2u)=u^p\quad\text{in }\R^n,\qquad -D^2u\in\overline{\Gamma_k},\qquad u\geq 0, \] where \(2\leq k<\frac{n}{2}\) and $p>0$. Let $p_- = \frac{nk}{n-2k}$ and the critical Hessian–Sobolev exponent $p_* = \frac{(n+2)k}{n-2k}$. Phuc and Verbitsky proved nonexistence of positive solutions for \(k<p\leq p_-\), while Ou subsequently covered the cases \(p\in(0,k]\). We close this gap and prove the optimal Liouville theorem for any \(p_-<p<p_*\): any nonnegative \(C^2\) entire solution must be identically zero. We also prove the optimal Liouville theorem for nonnegative locally bounded Hessian-measure weak solutions. This identifies the critical exponent \(p_*\) as the sharp Liouville threshold, since radial positive solutions exist for \(p\geq p_*\). For the critical case \(p=p_*\), we prove that every nontrivial nonnegative \(C^2\) entire solution is a Hessian–Sobolev bubble for every \(n>2k\), without any additional assumption. For the limiting case \(n=2k\), we classify finite-mass solutions to the $\frac{n}{2}$-Hessian Liouville equation under a proper asymptotic condition $u(x)\rightarrow-\infty$ as $|x|\rightarrow\infty$. In particular, we provide the fully nonlinear counterparts of the classical Liouville and classification theorems of Gidas–Spruck, Gidas–Ni–Nirenberg, and Caffarelli–Gidas–Spruck.

We prove here several tightness results for such quantum inequalities as 1) the comparison of operator norm and product norm of $d$-local hamiltonians, 2) Bohnenblust–Hille inequality for $d$-local hamiltonians, 3) for quantum Fourier entropy-influence conjecture. We give a deterministic prove of the sharpness of Bohnenblust–Hille exponent. We also discuss the quantum Aaronson–Ambainis conjecture in a special case of anti-commuting Pauli strings.

We study the three-dimensional Vlasov–Hartree system with Coulomb and inverse-power interactions. For small, regular, localized data, we prove global well-posedness and determine long-time dynamics. The main result concerns the Coulomb case, where the coupling persists at leading order through reciprocal asymptotic corrections: the fermionic distribution scatters along logarithmically corrected free characteristics set by the asymptotic bosonic profile, while the bosonic wave gains a logarithmic phase set by the asymptotic fermionic density. We also treat longer- and shorter-range interactions. Our proof combines Hamiltonian methods for the Vlasov equation with purely physical-space vector-field methods for dispersive equations. For stronger long-range interactions, we introduce a symplectic transformation and a regularized effective-phase cancellation, which remove leading nonintegrable interactions while preserving Hamiltonian structure and avoiding derivative loss.

We study positive $k$-peak solutions of the semiclassical fractional Schrödinger equation $\varepsilon^{2s}(-\Delta)^s u+V(x)u=u^p$ in $\mathbb{R}^N$, concentrating at different nondegenerate critical points $\xi_1^0,\ldots,\xi_k^0$ of $V$. For every such family satisfying the natural energy quantization, we determine the complete low spectrum of the linearized operator. The first $k$ eigenvalues remain uniformly negative, the next $kN$ eigenvalues are of order $\varepsilon^2$ and are governed by the Hessians $D^2V(\xi_j^0)$, while the remaining spectrum is uniformly separated from zero. Consequently, the Morse index equals $k$ plus the total number of negative eigenvalues of these Hessians, and every such solution is nondegenerate. Combining a unique modulation parametrization with a Leray–Schauder degree computation, we further prove that, for all sufficiently small $\varepsilon$, the prescribed concentrating class contains exactly one positive solution. The result applies to the whole energy-quantized class, not only to a particular solution.

The Calabi-Yau conjectures for complete minimal hypersurfaces $\Sigma^{n}\subset \mathbb{R}^{n+1}$ for $n\geq 2$ ask whether a complete minimal hypersurface must be unbounded, and more strongly whether it must be proper. In this work, we resolve this conjecture for complete, connected, embedded minimal hypersurfaces $ \Sigma^3 \subset \mathbb{R}^4$ with bounded second fundamental form and finite second Betti number $b_2(\Sigma;\mathbb{Z}_2)<\infty$.

We study internal controllability of Schrödinger equations on tori with controls acting only through their real parts. This real-part constraint leads to a real-linear control problem for which full null controllability may fail. We characterize the maximal null-controllable subspace and show that the uncontrollable directions are precisely given by the purely imaginary stationary modes. These results provide an explicit description of controllable and uncontrollable directions for Schrödinger equations with real-part controls. The characterization is obtained through observability inequalities with sharp stationary correction terms. For cylindrical open control regions, we prove such an estimate by a real-part compactness–uniqueness argument. For admissible measurable product control regions in the shifted free case, we establish the corresponding estimate for rough spacetime control sets. The proof combines complex-valued observability inputs with a double-spectrum analysis arising from the coupling of the forward and backward Schrödinger evolutions in the real-part observation.

We construct a $4$-parameter family of properly embedded genus-zero surfaces with at most two boundary components in a Riemannian $3$-ball. The construction is a free boundary analog of the canonical $5$-parameter family of surfaces on the $3$-sphere by Marques-Neves in the proof of Willmore conjecture. In the Euclidean unit ball, this family realizes the critical catenoid as a Simon-Smith min-max limit. As applications, we give an upper bound of Almgren-Pitts $4$-width $\omega_4(\mathbb{B}^3)$ by the area of the critical catenoid. Moreover, we prove the existence of at least three free boundary minimal annuli in any compact $3$-manifold with nonnegative Ricci curvature and strictly convex boundary.