Spatiotemporal model for the progression of transgressive dunes

Transgressive dune fields, which are active sand areas surrounded by vegetation, exist on many coasts. In some regions like in Fraser Island in Australia, small dunes shrink while large ones grow, although both experience the same climatic conditions. We propose a general mathematical model for the spatiotemporal dynamics of vegetation cover on sand dunes and focus on the dynamics of transgressive dunes. Among other possibilities, the model predicts growth parallel to the wind with shrinkage perpendicular to the wind, where, depending on geometry and size, a transgressive dune can initially grow although eventually shrink. The larger is the initial area the slower its stabilization process. The model’s predictions are supported by field observations from Fraser Island in Australia.     

Rigidity Theorems for Complete Sasakian Manifolds with Constant Pseudo-Hermitian Scalar Curvature

The orthogonal decomposition of the Webster curvature provides us a way to characterize some canonical metrics on a pseudo-Hermitian manifold. We derive some subelliptic differential inequalities from the Weitzenböck formulas for the traceless pseudo-Hermitian Ricci tensor of Sasakian manifolds with constant pseudo-Hermitian scalar curvature and the Chern–Moser tensor of the Sasakian pseudo-Einstein manifolds, respectively. By means of either subelliptic estimates or maximum principle, some rigidity theorems are established to characterize Sasakian pseudo-Einstein manifolds among Sasakian manifolds with constant pseudo-Hermitian scalar curvature and Sasakian space forms among Sasakian pseudo-Einstein manifolds, respectively.     

Why do active and stabilized dunes coexist under the same climatic conditions?

Sand dunes can be active (mobile) or stable, mainly as a function of vegetation cover and wind power. However, there exists as yet unexplained evidence for the coexistence of bare mobile dunes and vegetated stabilized dunes under the same climatic conditions. We propose a model for dune vegetation cover driven by wind power that exhibits bistabilty and hysteresis with respect to the wind power. For intermediate wind power, mobile and stabilized dunes can coexist, whereas for low (or high) wind power they can be fixed (or mobile). Climatic change or human intervention can turn active dunes into stable ones and vice versa; our model predicts that prolonged droughts with stronger winds can result in dune reactivation.     

Monotonicity formulae and Liouville theorems of harmonic maps with potential

We use the stress–energy tensor to obtain the monotonicity formula and vanishing theorems for harmonic maps with potential H$H$ under some conditions on H$H$. We also obtain the unique constant solution of the constant Dirichlet boundary value problem on some starlike domain for harmonic maps with potential.     

Liouville Theorems for F-Harmonic Maps and Their Applications

We prove several Liouville theorems for F-harmonic maps from some complete Riemannian manifolds by assuming some conditions on the Hessian of the distance function, the degrees of F(t) and the asymptotic behavior of the maps at infinity. In particular, the results can be applied to F-harmonic maps from some pinched manifolds, and can deduce a Bernstein type result for an entire minimal graph.     

On the structure of submanifolds in the hyperbolic space

Monatshefte für Mathematik - Let $$M^n$$ be a complete submanifold in the hyperbolic space $$\mathbb {H}^{n+m}$$ . We show the vanishing of the Betti numbers $$\beta _p(M)$$ , $$1 \le p \le...     

Localized structures in dryland vegetation: forms and functions

Vegetation patches in drylands are localized structures of biomass and water. We study these structures using a mathematical modeling approach that captures biomass-water feedbacks. Biomass-water structures are found to differ in their spatial forms and ecological functions, depending on species type, soil conditions, precipitation range, and other environmental factors. Asymptotic spot structures can destabilize to form ring structures, expanding in the radial direction, or crescent structures, migrating uphill. Stable spot structures can differ in their soil-water distributions, forming water-enriched patches or water-deprived patches. The various biomass-water structures are expected to function differently in the context of a plant community, forming landscapes of varying species diversity.     

关于局部对称空间中极小子流形的一个Ricci曲率pinching定理

研究局部对称空间中具有正Ricci曲率的完备极小子流形,得到了关于子流形Ricci曲率的一个pinching定理,把Norio Ejiri的结论从外围空间为球空间推广到局部对称空间中。