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The volume spectrum of a compact Riemannian manifold is a sequence of critical values for the area functional, defined in analogy with the Laplace spectrum by Gromov. In this paper we prove that the canonical metric on the two-dimensional projective plane is determined modulo isometries by its volume spectrum. We also prove that the surface Zoll metrics on the three-dimensional sphere are characterized by the equality of the spherical area widths. These widths generalize to the surface case the Lusternik-Schnirelmann lengths of closed geodesics. We prove a new sharp area systolic inequality for metrics on the three-dimensional projective space., Comment: 55 pages

In this work, eddy diffusivity is derived from the energy spectra for the stable and convective regimes in the planetary boundary layer. The energy spectra are obtained from a spectral model for the inertial subrange that considers the anomalous behavior of turbulence. This spectrum is expressed as a function of the Hausdorff fractal dimension. The diffusivity eddies are employed in a classical Eulerian dispersion model, where the derivatives are of integer order and in fractional dispersion model, where the derivatives are of fractional order. The eddy diffusivity proposed considers the anomalous nature of geophysical turbulent flow. The results obtained with the fractional and classic dispersion models using the eddy diffusivity proposed is satisfactory when compared with the experimental data of the Prairie Grass and Hanford experiments in a stable regime, and the Copenhagen experiment in a convective regime.

Animals chain movements into long-lived motor strategies, exhibiting variability across scales that reflects the interplay between internal states and environmental cues. To reveal structure in such variability, we build Markov models of movement sequences that bridges across time scales and enables a quantitative comparison of behavioral phenotypes among individuals. Applied to larval zebrafish responding to diverse sensory cues, we uncover a hierarchy of long-lived motor strategies, dominated by changes in orientation distinguishing cruising versus wandering strategies. Environmental cues induce preferences along these modes at the population level: while fish cruise in the light, they wander in response to aversive stimuli, or in search for appetitive prey. As our method encodes the behavioral dynamics of each individual fish in the transitions among coarse-grained motor strategies, we use it to uncover a hierarchical structure in the phenotypic variability that reflects exploration-exploitation trade-offs. Across a wide range of sensory cues, a major source of variation among fish is driven by prior and/or immediate exposure to prey that induces exploitation phenotypes. A large degree of variability that is not explained by environmental cues unravels motivational states that override the sensory context to induce contrasting exploration-exploitation phenotypes. Altogether, by extracting the timescales of motor strategies deployed during navigation, our approach exposes structure among individuals and reveals internal states tuned by prior experience., Comment: 31 pages, 7 main figures, 1 supplementary table and 8 supplementary figures

In this paper, we describe the intersection between geodesic and conformal currents on closed hyperbolic three-manifolds. We use this to prove some sharp bounds which involve the Liouville entropy of a negatively curved metric, the minimal surface entropy, and the area ratio. Using these ideas we also give a new proof of the Mostow Rigidity Theorem in the three-dimensional case., Comment: 35 pages

In this paper we construct smooth Riemannian metrics on the sphere which admit smooth Zoll families of minimal hypersurfaces. This generalizes a theorem of Guillemin for the case of geodesics. The proof uses the Nash-Moser Inverse Function Theorem in the tame maps setting of Hamilton. This answers a question of Yau on perturbations of minimal hypersurfaces in positive Ricci curvature. We also consider the case of the projective space and characterize those metrics on the sphere with minimal equators., Comment: 75 pages

Silica optical microspheres often exhibit ultra-high quality factors, yet, their group velocity dispersion, which is crucial for nonlinear optics applications, can only be coarsely tuned. We experimentally demonstrate that group-velocity dispersion of a silica microsphere can be engineered by coating it with conformal nanometric layers of alumina, yet preserving its ultra-high optical quality factors (\num{\sim e7}) at telecom wavelengths. Using the atomic layer deposition technique for the dielectric coating, which ensures nm-level thickness control, we not only achieve a fine dispersion tailoring but also maintain a low surface roughness and material absorption to ensure a low optical loss. Numerical simulations supporting our experimental results show that the alumina layer thickness is a promising technique for precise tuning of group-velocity dispersion. As an application we demonstrate the generation of Kerr optical frequency combs, showing that the alumina coatings can also sustain the high optical intensities necessary for nonlinear optical phenomena., Comment: main text: 7 pages, 4 figures; supplementary info: 8 pages

In this article we prove the strong Morse inequalities for the area functional in codimension one, assuming that the ambient dimension satisfies $3 \leq (n + 1) \leq 7$, in both the closed and the boundary cases., Comment: 29 pages

We introduced an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature less than or equal -1, by the hyperbolic metric., Comment: 27 pages

We show that strictly stable components of Allen-Cahn minimal hypersurfaces always occur with multiplicity one. We also establish the uniqueness of solutions converging to nondegenerate hypersurfaces with multiplicity one. Our results work in all dimensions and without variational assumptions on the Allen-Cahn solutions., Comment: 29 pages. We have improved our main result to a local version. Comments are welcome