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Abstract. We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finiteness, polynomial volume growth and properness of an immersed self-shrinker in Euclidean space.

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Let (M, g, e −f dμ) be a complete metric measure space with Bakry-Emery Ricci curvature bounded below by a positive constant. We prove that in M there is no complete two-sided L f -stable immersed f -minimal hypersurface with finite weighted volume. Further, if M is a 3-manifold, we prove a smooth compactness theorem for the space of complete embedded f -minimal surfaces in M with the uniform upper bounds of genus and weighted volume, which generalizes the compactness theorem for complete self-shrinkers in R 3 by Colding-Minicozzi.

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In this paper, first we study a complete smooth metric measure space (M n , g, e −f dv) with the (∞)-Bakry-Émery Ricci curvature Ric f ≥ a 2 g for some positive constant a. It is known that the spectrum of the drifted Laplacian ∆ f for M is discrete and the first nonzero eigenvalue of ∆ f has lower bound a 2 . We prove that if the lower bound a 2 is achieved with multiplicity k ≥ 1, then k ≤ n, M is isometric to Σ n−k × R k for some complete (n − k)-dimensional manifold Σ and by passing an isometry, (M n , g, e −f dv) must split off a gradient shrinking Ricci soliton (R k , gcan, a 4 |t| 2 ), t ∈ R k . This result has an application to gradient shrinking Ricci solitons. Secondly, we study the drifted Laplacian L for properly immersed self-shrinkers in the Euclidean space R n+p , p ≥ 1 and show the discreteness of the spectrum of L and a logarithmic Sobolev inequality.

Messenger RNA (mRNA) vaccines are being used to combat the spread of COVID-19 (refs. 1–3), but they still exhibit critical limitations caused by mRNA instability and degradation, which are major obstacles for the storage, distribution and efficacy of the vaccine products4. Increasing secondary structure lengthens mRNA half-life, which, together with optimal codons, improves protein expression5. Therefore, a principled mRNA design algorithm must optimize both structural stability and codon usage. However, owing to synonymous codons, the mRNA design space is prohibitively large—for example, there are around 2.4 × 10632 candidate mRNA sequences for the SARS-CoV-2 spike protein. This poses insurmountable computational challenges. Here we provide a simple and unexpected solution using the classical concept of lattice parsing in computational linguistics, where finding the optimal mRNA sequence is analogous to identifying the most likely sentence among similar-sounding alternatives6. Our algorithm LinearDesign finds an optimal mRNA design for the spike protein in just 11 minutes, and can concurrently optimize stability and codon usage. LinearDesign substantially improves mRNA half-life and protein expression, and profoundly increases antibody titre by up to 128 times in mice compared to the codon-optimization benchmark on mRNA vaccines for COVID-19 and varicella-zoster virus. This result reveals the great potential of principled mRNA design and enables the exploration of previously unreachable but highly stable and efficient designs. Our work is a timely tool for vaccines and other mRNA-based medicines encoding therapeutic proteins such as monoclonal antibodies and anti-cancer drugs7,8.

Let Ω be a bounded domain with convex boundary in a complete noncompact Riemannian manifold with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a lower bound of the first eigenvalue of the weighted Laplacian for closed embedded f -minimal hypersurfaces contained in Ω. Using this estimate, we prove a compactness theorem for the space of closed embedded fminimal surfaces with the uniform upper bounds of genus and diameter in a complete 3-manifold with Bakry-Émery Ricci curvature bounded below by a positive constant and admitting an exhaustion by bounded domains with convex boundary. 1 2 g or equivalently Ric f = 1 2 . We refer to [2], a survey of this topic where some compact and noncompact examples are explained. Even though the asymptotic growth of the potential function f of a noncompact shrinking gradient Ricci soliton is close to that of Gaussian shrinking soliton [3], both geometry and topology can be quite different. We may consider f -minimal hypersurfaces in a shrinking gradient Ricci soliton. For instance, a self shrinker in R n+1 can be viewed as an f -minimal hypersurface in a Gauss shrinking soliton (R n+1 , g can , |x| 2 4 ). There are other examples of f -minimal hypersurfaces. Let M be the hyperbolic space H n+1 (−1). Let r denote the distance function from a fixed point p ∈ M and f (x) = nar 2 (x), where a > 0 is a constant. Then Ric f ≥ n(2a − 1) and the geodesic sphere of radius r centered at p in H n+1 (−1) is an f -minimal hypersurface if it satisfies 2ar = coth r.An f -minimal hypersurface Σ has two aspects to view. One is that Σ is fminimal if and only if Σ is a critical point of the weighted volume functional e −f dσ, where dσ is the volume element of Σ. Another one is that Σ is fminimal if and only if Σ is minimal in the new conformal metricg = e − 2f n g (see Section 2). f -minimal hypersurfaces have been studied before, even more general stationary hypersurfaces for parametric elliptic functionals, see for instance the work of White [21] and .In this paper, we will first estimate the lower bound of the first eigenvalue of the weighted Laplacian ∆ f = ∆ − ∇f, ∇· for closed (i.e. compact and without boundary) embedded f -minimal hypersurfaces in a complete metric measure space (M, g, e −f dµ). Subsequently using the eigenvalue estimate,