Geometric Evolution of the Chongce Glacier during 1970–2020, Detected by Multi-Source Satellite Observations

Glacier surge, which causes a quick movement of ice mass from high to low elevation, is closely associated to the glacial hazards of debris flows and glacial lake outburst floods. Over the West Kunlun Shan, surge events have been detected for some glaciers, however, the characteristics (e.g., the active phase) of the identified surge-type glaciers are not fully understood due to the paucity of long-term observations of glacier changes. In this study, we investigated the geometric evolution of the Chongce Glacier (a surge-type glacier) over the past five decades. Glacier elevation changes were observed by comparing topographic data from different times. Surface velocity and terminus position were derived using a cross-correlation algorithm and band ratio method, respectively. A decreasing rate of glacier surface thinning was found for the Chongce Glacier during the studied period. Glacier elevation changes of −0.46 ± 0.12, −0.12 ± 0.05, and 0.27 ± 0.11 m yr−1 were estimated for the periods of 1970–2000, 2000–2012, and 2012–2018, respectively. Moreover, this glacier experienced obvious surface lowering over the terminus zone and clear surface thickening over the upper zone during 1970–2000, and the opposite during 2000–2018. Surface velocity of the Chongce Glacier was less than 300 m yr−1 in 1990–1993, and then quickly increased to more than 1000 m yr−1 between 1994 and 1998, and dropped to less than 50 m yr−1 in 1999–2020. Over the past five decades, the Chongce Glacier generally experienced a slight retreat, except for a terminus advance from 1995 to 1999. According to the spatial pattern of glacier elevation changes in 1970–2000 and the long-term changes of glacier velocity and terminus position, the recent surge event at the Chongce Glacier likely initiated in winter 1993 and terminated in winter 1998. Furthermore, the start date, end date, and duration of the active phase indicate that the detected surge event was likely triggered by a thermal mechanism.

On the Electromagnetic Vacuum Origins of Dark Energy, Dark Matter, and Astrophysical Jets

We use a semiclassical version of the Nexus paradigm of quantum gravity in which the quantum vacuum at large scales is dominated by the second quantized electromagnetic field to demonstrate that a virtual photon field can affect the geometric evolution of Einstein manifolds or Ricci solitons. This phenomenon offers a cogent explanation of the origins of astrophysical jets, the cosmological constant, and a means of detecting galactic dark matter.

Stable approximations for axisymmetric Willmore flow for closed and open surfaces

For a hypersurface in $\mathbb R^3$, Willmore flow is defined as the $L^2$--gradient flow of the classical Willmore energy: the integral of the squared mean curvature. This geometric evolution law is of interest in differential geometry, image reconstruction and mathematical biology. In this paper, we propose novel numerical approximations for the Willmore flow of axisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants we prove a stability result. We consider both closed surfaces, and surfaces with a boundary. In the latter case, we carefully derive weak formulations of suitable boundary conditions. Furthermore, we consider many generalizations of the classical Willmore energy, particularly those that play a role in the study of biomembranes. In the generalized models we include spontaneous curvature and area difference elasticity (ADE) effects, Gaussian curvature and line energy contributions. Several numerical experiments demonstrate the efficiency and robustness of our developed numerical methods.

Off-diagonal cosmological solutions in emergent gravity theories and Grigory Perelman entropy for geometric flows

We develop an approach to the theory of relativistic geometric flows and emergent gravity defined by entropy functionals and related statistical thermodynamics models. Nonholonomic deformations of G. Perelman’s functionals and related entropic values used for deriving relativistic geometric evolution flow equations. For self-similar configurations, such equations describe generalized Ricci solitons defining modified Einstein equations. We analyse possible connections between relativistic models of nonholonomic Ricci flows and emergent modified gravity theories. We prove that corresponding systems of nonlinear partial differential equations, PDEs, for entropic flows and modified gravity posses certain general decoupling and integration properties. There are constructed new classes of exact and parametric solutions for nonstationary configurations and locally anisotropic cosmological metrics in modified gravity theories and general relativity. Such solutions describe scenarios of nonlinear geometric evolution and gravitational and matter field dynamics with pattern-forming and quasiperiodic structure and various space quasicrystal and deformed spacetime crystal models. We analyse new classes of generic off-diagonal solutions for entropic gravity theories and show how such solutions can be used for explaining structure formation in modern cosmology. Finally, we speculate why the approaches with Perelman–Lyapunov type functionals are more general or complementary to the constructions elaborated using the concept of Bekenstein–Hawking entropy.

Liquid crystals on deformable surfaces

Liquid crystals with molecules constrained to the tangent bundle of a curved surface show interesting phenomena resulting from the tight coupling of the elastic and bulk-free energies of the liquid crystal with geometric properties of the surface. We derive a thermodynamically consistent Landau-de Gennes-Helfrich model which considers the simultaneous relaxation of the Q -tensor field and the surface. The resulting system of tensor-valued surface partial differential equation and geometric evolution laws is numerically solved to tackle the rich dynamics of this system and to compute the resulting equilibrium shape. The results strongly depend on the intrinsic and extrinsic curvature contributions and lead to unexpected asymmetric shapes.

Structure-preserving discretizations of gradient flows for axisymmetric two-phase biomembranes

The form and evolution of multi-phase biomembranes are of fundamental importance in order to understand living systems. In order to describe these membranes, we consider a mathematical model based on a Canham–Helfrich–Evans two-phase elastic energy, which will lead to fourth-order geometric evolution problems involving highly nonlinear boundary conditions. We develop a parametric finite element method in an axisymmetric setting. Using a variational approach it is possible to derive weak formulations for the highly nonlinear boundary value problems such that energy decay laws, as well as conservation properties, hold for spatially discretized problems. We will prove these properties and show that the fully discretized schemes are well posed. Finally, several numerical computations demonstrate that the numerical method can be used to compute complex, experimentally observed two-phase biomembranes.