In this paper, we compute the Morse index of a free boundary minimal submanifold from data of two simpler problems. The first is the fixed boundary problem and the second is concered with the Dirichlet-to-Neumann map associated with the Jacobi operator. As an application, we show that the Morse index of a free boundary minimal annulus is equal to 4 if and only if it is the critical catenoid.

Abstract. This paper derives new identities for the Weyl tensor on a gradient Ricci soliton, particularly in dimension four. First, we prove a Bochner-Weitzenböck type formula for the norm of the self-dual Weyl tensor and discuss its applications, including connections between geometry and topology. In the second part, we are concerned with the interaction of different components of Riemannian curvature and (gradient and Hessian of) the soliton potential function. The Weyl tensor arises naturally in these investigations. Applications here are rigidity results.

Data from social media systems are being actively mined for trends and patterns of interests. Problems such as sentiment and opinion mining, and prediction of election outcomes have become tremendously popular due to the unprecedented availability of social interactivity data of different types. An important angle that has not yet been explored is to estimate beliefs from posts made on social media. We propose that social media can be used to monitor the level of belief, disbelief and doubt related to specific propositions. Inspired by efforts in disease surveillance using social media we coin the term belief surveillance for this function. We propose a novel methodological framework for belief surveillance using Twitter. Our method may be used to gauge belief on any proposition as long as it is specifiable in a form that we call probes. We present our belief estimates for 32 probes some of which represent factual information, others represent false information and the remaining represent debatable propositions. Finally, we provide preliminary evidence suggesting that off-the-shelf classifiers may be used to automatically estimate belief.

Entering the UN Decade on Ecosystem Restoration, interventions referred to as nature-based solutions (NBS) are at the forefront of the sustainability discourse. While applied in urban, natural forest or wetland ecosystems, they are underutilized in agricultural landscapes. This paper presents a technical framework to characterise NBS in agricultural systems. NBS in the agriculture sector is proposed as “the use of natural processes or elements to improve ecosystem functions of environments and landscapes affected by agricultural practices, and to enhance livelihoods and other social and cultural functions, over various temporal and spatial scales.” The framework emerges from a review of 188 peer-reviewed articles on NBS and green infrastructure published between 2015 and 2019 and three international expert consultations organized in 2019–2020. The framework establishes four essential functions for NBS in agriculture: 1) Sustainable practices — with a focus on production; 2) Green Infrastructure — mainly for engineering purposes such as water and soil, and slope stabilization; 3) Amelioration — for restoration of conditions for plants, water, soil or air and climate change mitigation; and 4) Conservation — focusing on biodiversity and ecosystem connectivity. The framework connects the conventional divide between production and conservation to add functionality, purpose and scale in project design. The review confirmed limited evidence of NBS in agricultural systems particularly in developing country contexts, although specific technologies feature under other labels. Consultations indicated that wider adoption will require a phased approach to generate evidence, while integrating NBS in national and local policies and agricultural development strategies. The paper concludes with recommended actions required to facilitate such processes.

Functionals involving surface curvature are important across a range of scientific disciplines, and their extrema are representative of physically meaningful objects such as atomic lattices and biomembranes.Inspired in particular by the relationship of the Willmore energy to lipid bilayers, we consider a general functional depending on a surface and a symmetric combination of its principal curvatures, provided the surface is immersed in a 3-D space form of constant sectional curvature. We calculate the first and second variations of this functional, extending known results and providing computationally accessible expressions given entirely in terms of the basic geometric information found in the surface fundamental forms. Further, we motivate and introduce the p-Willmore energy functional, applying the stability criteria afforded by our calculations to prove a result about the p-Willmore energy of spheres. 1 2 ON THE VARIATION OF CURVATURE FUNCTIONALSHelfrich model for membrane energy per unit area is given by the functionalwhere H is the membrane mean curvature, K is its Gauss curvature, k, k c are some rigidity constants, and c 0 is a constant known as the "spontaneous curvature". Physically, this high dependence on curvature arises from hydrostatic pressure differences between the fluids internal and external to the membrane.Another noteworthy curvature functional is the bending energy, which quantifies how much (on average) a surface M deviates from being a round sphere. Specifically, the bending energy functional is defined aswhere k 0 is the sectional curvature of the ambient space. This type of energy was first considered by Sophie Germain in 1811 (see [12]) as a model for the bending energy of a thin plate. In particular, she suggested that the bending energy be measured by an integral over the plate surface, taking as integrand some symmetric and even-degree polynomial in the principal curvatures. Note that the functional (2) is one of the simplest models of this kind.Remark. The bending energy also arises in the field of computer vision, where changes in surface curvature are used to simulate natural movement. On the other hand, it is known to these scientists as the surface torsion (see [13]) due to how it measures the change in normal curvature of the surface.From a mathematical perspective, both the Helfrich energy and the bending energy are closely related to the conformally invariant (see [14]) Willmore energy popularized in [15], which is defined as(3) W(M ) :=