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It is shown that nonlinear electrodynamics of the Born–Infeld theory type may be exploited to shed insight into a few fundamental problems in theoretical physics, including rendering electromagnetic asymmetry to energetically exclude magnetic monopoles, achieving finite electromagnetic energy to relegate curvature singularities of charged black holes, and providing theoretical interpretation of equations of state of cosmic fluids via k-essence cosmology. Also discussed are some nonlinear differential equation problems.

Similar to the Schwarzschild coordinates for spherical black holes, the Baldwin, Jeffery and Rosen (BJR) coordinates for plane gravitational waves are often singular, and extensions beyond such singularities are necessary, before studying asymptotic properties of such spacetimes at the null infinity of the plane, on which the gravitational waves propagate. The latter is closely related to the studies of memory effects and soft graviton theorems. In this paper, we point out that in the BJR coordinates all the spacetimes are singular physically at the focused point u = u s , except for the two cases: (1) α = 1 / 2 , ∀ χ n ; and (2) α = 1 , χ i = 0 (i = 1 , 2 , 3) , where χ n are the coefficients in the expansion χ ≡ det g ab 1 / 4 = u - u s α ∑ n = 0 ∞ χ n u - u s n with χ 0 ≠ 0 , the constant α ∈ (0 , 1 ] characterizes the strength of the singularities, and g ab denotes the reduced metric on the two-dimensional plane orthogonal to the propagation direction of the wave. Therefore, the hypersurfaces u = u s already represent the boundaries of such spacetimes, and the null infinity does not belong to them. As a result, they cannot be used to study properties of plane gravitational waves at null infinities, including memory effects and soft graviton theorems. [ABSTRACT FROM AUTHOR]

Abstract We study the regularity of the distance function to the boundary of a domain in R 2 , with respect to some asymmetric norms. We allow the boundary of the domain to have corners. We obtain an explicit formula for the second derivative of these distance functions. Furthermore, we study a generalized notion of the ridge of a domain, which is the set of singularities of a distance function to the boundary of the domain. We completely characterize the ridge by using a generalized notion of curvature. [ABSTRACT FROM AUTHOR]

A framed surface is a smooth surface in the Euclidean space with a moving frame. The framed surfaces may have singularities. We treat smooth surfaces with singular points, that is, singular surfaces more directly. By using the moving frame, the basic invariants and curvatures of the framed surface are introduced. Then we show that the existence and uniqueness for the basic invariants of the framed surfaces. We give properties of framed surfaces and typical examples. Moreover, we construct framed surfaces as one-parameter families of Legendre curves along framed curves. We give a criteria for singularities of framed surfaces by using the curvature of Legendre curves and framed curves. [ABSTRACT FROM AUTHOR]

Abstract A spacetime denotes a pure radiation field if its energy–momentum tensor represents a situation in which all the energy is transported in one direction with the speed of light. In 1989 Wils and later in 1997 Ludwig and Edgar studied the physical properties of pure radiation metrics, which are conformally related to a vacuum spacetime. In the present paper we investigate the curvature properties of the pure radiation metrics presented by Ludwig and Edgar. It is shown that such a pure radiation spacetime is semisymmetric, Ricci simple, R -space by Venzi and its Ricci tensor is Riemann compatible. It is also proved that its conformal curvature 2-forms and Ricci 1-forms are recurrent. We also present a pure radiation type metric and evaluate its curvature properties along with the form of its energy–momentum tensor. It is interesting to note that such a pure radiation type metric is E i n (3) and 3-quasi-Einstein. We also find out the sufficient conditions for which this metric represents a generalized pp-wave, pure radiation and perfect fluid. Finally we made a comparison between the curvature properties of Ludwig and Edgar's pure radiation metric and pp-wave metrics. [ABSTRACT FROM AUTHOR]

In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time behavior of the solution and obtain the convergence result. In particular, we give full characterization of the limiting profiles in the radially symmetric setting. [ABSTRACT FROM AUTHOR]

Abstract We introduce a new version of a curvature-dimension inequality for non-negative curvature. We use this inequality to prove a logarithmic Li–Yau inequality on finite graphs. To formulate this inequality, we introduce a non-linear variant of the calculus of Bakry and Émery. In the case of manifolds, the new calculus and the new curvature-dimension inequality coincide with the common ones. In the case of graphs, they coincide in a limit. In this sense, the new curvature-dimension inequality gives a more general concept of curvature on graphs and on manifolds. We show that Ricci-flat graphs have a non-negative curvature in this sense. Moreover, a variety of non-logarithmic Li–Yau type gradient estimates can be obtained by using the new Bakry–Émery type calculus. Furthermore, we use these Li–Yau inequalities to derive Harnack inequalities on graphs. [ABSTRACT FROM AUTHOR]

In this paper, using the Gibbs dividing surface method, the formula that determines the curvature-dependent surface tension in a system with two phases is deduced. The well-known Tolman formula is a special case for this formula. The problem of a sessile droplet is considered. The Bashforth-Adams equation analogue in view of the curvaturedependent surface tension is obtained and the numerical solution of the equation is carried out. We show that if the droplet size is not so large compared to the thickness of the surface layer (micro- or nanodroplets), the dependence of the surface tension on the curvature is very important. [ABSTRACT FROM AUTHOR]

κ-solutions are very important to the study of Ricci flow since they serve as the finite-time singularity models. With the help of his profound understanding of κ-solutions, Perelman [11] made the major breakthrough in Hamilton's program. However, three-dimensional κ-solutions are not yet classified until this day. We prove a classification result assuming a Type I curvature bound. [ABSTRACT FROM AUTHOR]

We investigate the relation between two types of space curves, the Mannheim curves and constant-pitch curves and primarily explicate a method of deriving Mannheim curves and constant-pitch curves from each other by means of a suitable deformation of a space curve. We define a "radius" function and a "pitch" function for any arbitrary regular space curve and use these to characterize the two classes of curves. A few non-trivial examples of both Mannheim and constant pitch curves are discussed. The geometric nature of Mannheim curves is established by using the notion of osculating helices. The Frenet–Serret motion of a rigid body in theoretical kinematics is studied for the special case of a Mannheim curve and the axodes in this case are deduced. In particular, we show that the fixed axode is developable if and only if the motion trajectory is a Mannheim curve. [ABSTRACT FROM AUTHOR]

Non-perturbative quantum gravity prevents the formation of curvature singularities and may allow black holes to decay with a lifetime shorter than evaporation time. This, in connection with the existence of primordial black holes, could open a new window for quantum-gravity phenomenology. I discuss the possibility of observing astrophysical emissions from the explosion of old black holes in the radio and in the gamma wavelengths. These emissions can be discriminated from other astrophysical sources because of a peculiar way the emitted wavelength scales with the distance. The spectrum of the diffuse radiation produced by those objects presents a peculiar distortion due to this scaling. [ABSTRACT FROM AUTHOR]

The existence of spacetime singularities is one of the biggest problems of nowadays physics. According to Penrose, each physical singularity should be covered by a “cosmic censor” which prevents any external observer from perceiving their existence. However, classical models describing the gravitational collapse usually results in strong curvature singularities, which can also remain “naked” for a finite amount of advanced time. This proceedings studies the modifications induced by asymptotically safe gravity on the gravitational collapse of generic Vaidya spacetimes. It will be shown that, for any possible choice of the mass function, quantum gravity makes the internal singularity gravitationally weak, thus allowing a continuous extension of the spacetime beyond the singularity. [ABSTRACT FROM AUTHOR]

We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski 3-space. [ABSTRACT FROM AUTHOR]

In this paper we consider the evolution of boundaries of sets by a fractional mean curvature flow. We show that for any dimension n ≥ 2, there exist embedded hypersurfaces in Rn which develop a singularity without shrinking to a point. Such examples are well known for the classical mean curvature flow for n ≥ 3. Interestingly, when n = 2, our result provides instead a counterexample in the nonlocal framework to the well-known Grayson’s Theorem, which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point. The essential step in our construction is an estimate which ensures that a suitably small perturbation of a thin strip has positive fractional curvature at every boundary point. [ABSTRACT FROM AUTHOR]

We consider statistical submanifolds of Hessian manifolds of constant Hessian curvature. For such submanifolds we establish a Euler inequality and a Chen-Ricci inequality with respect to a sectional curvature of the ambient Hessian manifold. [ABSTRACT FROM AUTHOR]

The simplest way to have birth of surfaces is through transitions in the fibres of a function f with a Morse singularity of index 0 or 3. It is natural to seek to understand the geometry of newly born surfaces. We consider here the question of finding how many umbilics are on a newly born surface. We show that newly born surfaces in the Euclidean 3-space have exactly 4 umbilic points all of type lemon, provided that the Hessian of f at the singular point has pairwise distinct eigenvalues. This is true in both cases when f is an analytic or a smooth germ. When only two of such eigenvalues are equal, the number of umbilic points is either 2, 4, 6 or 8 when f is an analytic or a generic smooth germ. The same results holds for newly born surfaces in the Minkowski 3-space. In that case when the two eigenvalues associated to the two spacelike eigenvectors are distinct we get exactly 4 umbilic points all of type lemon. If they are equal, the number of umbilic points is either 2, 4, 6 or 8. [ABSTRACT FROM AUTHOR]

We solve a $$1+5$$ -dimensional cylindrical gravitational-wave solution of the Einstein equation, in which there are two curvature singularities. Then we show that one of the curvature singularities can be removed by an extension of the spacetime. The result exemplifies that the curvature singularity is not always a spacetime singularity; in other words, the curvature singularity cannot serve as a criterion for spacetime singularities. [ABSTRACT FROM AUTHOR]

We show that a Morse-type potential simulates an analytic solution for the highly nonlinear global monopole field equation in three and higher-dimensional flat spacetimes. Owing to the fact that in the flat space limit the similar equation remains intact, we wish to borrow the curved space terminology of global monopole also in flat spacetime. This may provide a compelling example that can be used effectively in different nonlinear theories such as flat space as well as in curved spacetimes. [ABSTRACT FROM AUTHOR]

We study a Neumann problem with the Hardy–Sobolev nonlinearity. In boundary singularity case, the impact of the mean curvature at singularity on existence of least-energy solution is well known. Existence and nonexistence of least-energy solution is studied by Hashizume (2017) except for lower dimension case. In this paper, we improve this previous work. More precisely, we study four dimensional case and show existence of minimizer in critical case in some sense. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of almost regular (α,β)-metrics constructed by Shen called unicorn metrics. First, we prove that every unicorn metric with almost vanishing H-curvature is a Berwald metric. Then we show that every unicorn metric with almost vanishing Ξ-curvature reduces to a Berwald metric. [ABSTRACT FROM AUTHOR]

In this paper, we consider the semi-linear oblique boundary value problem for the prescribed mean curvature equation. We find a suitable auxiliary function and use the maximum principle to get the gradient estimate. As a consequence, we obtain the corresponding existence theorem for a class of mean curvature equations with semi-linear oblique derivative problems [ABSTRACT FROM AUTHOR]

For appropriate choices of the coupling constants, the equations of motion of Lovelock gravities up to order n in the Riemann tensor can be factorized such that the theories admit a single (A)dS vacuum. In this paper we construct two classes of exact rotating metrics in such critical Lovelock gravities of order n in d = 2 n + 1 dimensions. In one class, the n angular momenta in the n orthogonal spatial 2-planes are equal, and hence the metric is of cohomogeneity one. We construct these metrics in a Kerr–Schild form, but they can then be recast in terms of Boyer–Lindquist coordinates. The other class involves metrics with only a single non-vanishing angular momentum. Again we construct them in a Kerr–Schild form, but in this case it does not seem to be possible to recast them in Boyer–Lindquist form. Both classes of solutions have naked curvature singularities, arising because of the over rotation of the configurations. [ABSTRACT FROM AUTHOR]

A reconstruction algorithm for blade surface from less measured points of section curves is given based on B-spline surface interpolation. The less measured points are divided into different segments by the key geometric points and throat points which are defined according to design concepts. The segmentations are performed by different fitting algorithms with consideration of curvature continuity as their boundary condition to avoid flow disturbance. Finally, a high-quality reconstruction surface model is obtained by using the B-spline curve meshes constructed by paired points. The advantage of this algorithm is the simplicity and effectivity reconstruction of blade surface to ensure the aerodynamic performance. Moreover, the obtained paired points can be regarded as measured points to measure and reconstruct the blade surface directly. Experimental results show that the reconstruction blade surface is suitable for precisely representing blade, evaluating machining accuracy, and analyzing machining allowance. [ABSTRACT FROM AUTHOR]

Black hole solutions to the Einstein equations coupled with the Born–Infeld electromagnetism associated with generalized nonlinear electrodynamics, carrying both electric and magnetic charges, often referred to as dyonically charged black holes, of finite energies, are constructed. These solutions give rise to relegated curvature singularities at the center of the matter sources of the black holes and approach the classical Reissner–Nordström black hole asymptotically. • Black hole solutions to the Einstein equations coupled with the Born–Infeld electromagnetism associated with generalized nonlinear electrodynamics, carrying both electric and magnetic charges, often referred to as dyonically charged black holes, of finite energies, are constructed. • These solutions give rise to relegated curvature singularities at the center of the matter sources of the black holes and approach the classical Reissner–Nordstrom black hole asymptotically. [ABSTRACT FROM AUTHOR]

In this paper we give a new proof for two sharp inequalities involving generalized normalized δ-Casorati curvatures of a slant submanifold in a quaternionic space form. These inequalities were recently obtained in Lee and Vîlcu (Taiwan. J. Math. 19(3):691-702, 2015) using an optimization procedure by showing that a quadratic polynomial in the components of the second fundamental form is parabolic. The new proof is obtained analyzing a suitable constrained extremum problem on submanifold. [ABSTRACT FROM AUTHOR]

In this paper, we prove a generalization of Reilly's formula in [10]. We apply such general Reilly's formula to give alternative proofs of the Alexandrov's Theorem and the Heintze–Karcher inequality in the hemisphere and in the hyperbolic space. Moreover, we use the general Reilly's formula to prove a new Heintze–Karcher inequality for Riemannian manifolds with boundary and sectional curvature bounded below. [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider the mean curvature flow of surfaces in hyperbolic 3-space. First, we establish the isoperimetric inequality using the flow, provided the enclosed volume approaches zero at the final time. Second, we construct two singular examples of the flow. More precisely, there exists a torus which must develop a singularity in the flow before the volume it encloses decreases to zero. There also exists a topological sphere in the shape of dumbbell, which must develop a singularity in the flow before its area shrinks to zero. [ABSTRACT FROM AUTHOR]

We show that effective theories of matter that classically violate the null energy condition cannot be minimally coupled to Einstein gravity without being inconsistent with both string theory and black hole thermodynamics. We argue however that they could still be either non-minimally coupled or coupled to higher-curvature theories of gravity. [ABSTRACT FROM AUTHOR]

In this paper, we study the case of single field inflation within the framework of modified gravity theory where the gravity part has an arbitrary form f ( R ) . Via a conformal transformation, this case can be transformed into its Einstein frame where it looks like a two-field inflation model. However, due to the existence of the isocurvature modes in such a multi-degree-of-freedom (m.d.o.f.) system, the (curvature) perturbations are not equivalent in two frames, so despite of its convenience, it is illegal to treat the perturbations in its Einstein frame as the “real” ones as we always do for pure f ( R ) theory or single field with nonminimal coupling. Here by pulling the results of curvature perturbations back into its original Jordan frame, we show explicitly the power spectrum and spectral index of the perturbations in the Jordan frame, as well as how it differs from the Einstein frame. We also fit our results with the newest Planck data. Since there is large parameter space in these models, we show that it is easy to fit the data very well. [ABSTRACT FROM AUTHOR]

We derive a generalized Gross-Pitaevski (GP) equation for a Bose Einstein Condensate (BEC) immersed in a weak gravitational field starting from the covariant Complex Klein-Gordon field in a curved space-time. We compare it with the traditional GP equation where the gravitational field is added by hand as an external potential. We show that there is a small difference of order gz/ c between them that could be measured in the future using Bose-Einstein Condensates. This represents the next order correction to the Newtonian gravity in a curved space-time. Graphical abstract: [Figure not available: see fulltext.] [ABSTRACT FROM AUTHOR]

A constant curvature spherically symmetric vacuum collapse in f( R) gravity is investigated. It is found that even a vacuum collapse may lead to a future curvature singularity which may be covered by a horizon leading to a black hole depending on the initial profile of the geometry. [ABSTRACT FROM AUTHOR]

We study complete noncompact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds endowed with a timelike conformal vector field V. In this setting, by using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds, we establish new characterizations of totally umbilical hypersurfaces in terms of their higher order mean curvatures. For instance, supposing an appropriated restriction on the norm of the tangential component of the vector field V, we are able to show that such hypersurfaces must be totally umbilical provided that either some of their higher order mean curvatures are linearly related or one of them is constant. Applications to the so-called generalized Robertson-Walker spacetimes are given. In particular, we extend to the Lorentzian context a classical result due to Jellett . [ABSTRACT FROM AUTHOR]

We analyze the persistence of curvature singularities when analyzed using quantum theory. First, quantum test particles obeying the Klein-Gordon and Chandrasekhar-Dirac equation are used to probe the classical timelike naked singularity. We show that the classical singularity is felt even by our quantum probes. Next, we use loop quantization to resolve a singularity hidden beneath the horizon. The singularity is resolved in this case. [ABSTRACT FROM AUTHOR]

Based on the nuclear equation of state, the curvature correction term to the surface tension coefficient is calculated. Tolman's ... correction is shown to be sensitive to the Skyrme force parametrization. The temperature dependence of the Tolman length, important in heavy ion collision experiments, is derived. In the present approach the curvature term is related to the bulk properties of the nuclear matter through the equation of state. The results are compared with the existing theoretical calculations based on the Gibbs-Tolman formalism and with the theoretical predictions concerning its dependence on the interparticle distance. [ABSTRACT FROM AUTHOR]

With the objective to find a solution to the cosmological constant and the coincidence problems, we consider a cosmological model where the dark sectors are interacting together through a phenomenological decay law $$\dot{\rho }_\mathrm{\Lambda }=Q\rho _\mathrm{\Lambda }^n$$ in a FRW spacetime with spatial curvature. We show that the only value of $$n$$ for which the late-time matter energy density to dark energy density ratio ( $$r\mathrm{_m}=\rho _\mathrm{m}/\rho _\mathrm{\Lambda }$$ ) is constant (which could provide an explanation to the coincidence problem) is $$n=3/2$$ . For each value of $$Q$$ , there are two distinct solutions. One of them involves a spatial curvature approaching zero at late times ( $$\rho _\mathrm{k}\approx 0$$ ) and is stable when the interaction is weaker than a critical value $${Q_0=-\sqrt{32\pi G/c^2}}$$ . The other one allows for a non-negligible spatial curvature ( $$\rho _\mathrm{k}\not \approx 0$$ ) at late times and is stable when the interaction is stronger than $$Q_0$$ . We constrain the model parameters using various observational data (SNeIa, GRB, CMB, BAO, OHD). The limits obtained on the parameters exclude the regions where the cosmological constant problem is significantly ameliorated and do not allow for a completely satisfying explanation for the coincidence problem. [ABSTRACT FROM AUTHOR]

We study the semiclassical behavior of Lorentzian Engle–Pereira–Rovelli–Livine (EPRL) spinfoam model, by taking into account the sum over spins in the large spin regime. We also employ the method of stationary phase analysis with parameters and the so-called, almost analytic machinery, in order to find the asymptotic behavior of the contributions from all possible large spin configurations in the spinfoam model. The spins contributing the sum are written as Jf = λjf, where λ is a large parameter resulting in an asymptotic expansion via stationary phase approximation. The analysis shows that at least for the simplicial Lorentzian geometries (as spinfoam critical configurations), they contribute the leading order approximation of spinfoam amplitude only when their deficit angles satisfy mod . Our analysis results in a curvature expansion of the semiclassical low energy effective action from the spinfoam model, where the UV modifications of Einstein gravity appear as subleading high-curvature corrections. [ABSTRACT FROM AUTHOR]

Here we present an example of an axially symmetric spacetime, representing pure radiation, and admitting circular closed timelike curves (CTCs) on the $$z= \hbox {constant plane}$$ . The spacetime is regular everywhere, having no curvature singularities and is locally isometric to (non-vacuum) pp wave spacetimes. The stability of the CTCs under linear perturbations is studied and they are found to be stable from a calculation of the Lyapunov exponent for the deviation vector. We also demonstrate that the spacetime also admits non-circular CTCs which do not lie in this plane. A modification of the metric is also studied and we find that a region of this spacetime behaves like a time-machine where CTCs appear after a certain instant of time. [ABSTRACT FROM AUTHOR]

With a prescribed Coulomb-type energy-momentum tensor, an exact solution of the Einstein field equations over a nonsimply-connected manifold is presented. This spherically symmetric solution has neither curvature singularities nor closed timelike curves. It can be considered to be a regularization of the singular Reissner-Nordström solution over a simply-connected manifold. [ABSTRACT FROM AUTHOR]

A simple argument indicates that covariant loop gravity (spin foam theory) predicts a maximal acceleration and hence forbids the development of curvature singularities. This supports the results obtained for cosmology and black holes using canonical methods. [ABSTRACT FROM AUTHOR]

We study the thermodynamics and geometrothermodynamics of different black hole configurations in more than four spacetime dimensions. We use the response functions to find the conditions under which second order phase transitions occur in higher-dimensional static Reissner-Nordström and stationary Kerr black holes. Our results indicate that the equilibrium manifold of all these black hole configurations is in general curved and that curvature singularities appear exactly at those places where second order phase transitions occur. [ABSTRACT FROM AUTHOR]

The speed of fragmented picture identification depends on a large number of factors whose effects might change in time during an identification attempt. Using survival analysis and fixed fragmentation levels, previous research has shown that effects of complexity, fragment curvature, and time interact. Here, we study the effects of presentation duration and dynamic fragmentation levels. Fragmented object outlines were presented repetitively every 2.25 s, and at each presentation longer fragments were shown (possibly until closure). We recorded the lowest presentation number (minimum 1, maximum 10) that resulted in correct naming by the participants (N=84). Survival analysis was employed to investigate whether and when different factors like presentation duration, complexity, object category (natural vs. artifactual), symmetry, proximity, and fragment curvature influence correct identification. The results confirm and extend previous findings, and are interpreted within a dynamic, interactive processing framework. [ABSTRACT FROM AUTHOR]

To study the dynamic interplay between different component processes involved in the identification of fragmented object outlines, the authors used a discrete-identification paradigm in which the masked presentation duration of fragmented object outlines was repeatedly increased until correct naming occurred. Survival analysis was used to investigate whether and when different types of information--such as contour integration cues (proximity, collinearity, and fragment density), fragment properties (low vs. high curvature), stimulus complexity (global symmetry, number and saliency of the parts), and memory factors (natural vs. artifactual)--influenced the timing of identification. The results show that the importance of these different types of information can change over the time course of object identification, indicating so-called time-course contingencies. Most important, the straight segments of a contour played a larger role for complex outlines with high part saliency during early (bottom-up) grouping processes, whereas the curved segments of object outlines were more important during later (top-down) matching processes for simpler outlines with lower part saliency. This new insight can explain why different studies on shape-based object identification have produced seemingly contradictory results. [ABSTRACT FROM AUTHOR]

Let M be a Kähler-Einstein surface with positive scalar curvature. If the initial surface is sufficiently close to a holomorphic curve, we show that the mean curvature flow has a global solution and it converges to a holomorphic curve. [ABSTRACT FROM AUTHOR]

We introduce a model for Hermitian-holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular Hermitian structures along a normal crossing divisor. In the case of a projective curve, the cup product in cohomology is shown to correspond to a generalization of the Deligne pairing to line bundles with “good” Hermitian metrics in the sense of Mumford and others. A particular case is that of the tangent bundle of the curve twisted by the negative of the singularity divisor of a hyperbolic metric: its cup square (corrected by the total area) is shown to be a functional whose extrema are the metrics of constant negative curvature. [ABSTRACT FROM AUTHOR]