ON THE NONNEGATIVITY OF THE DIRICHLET ENERGY OF A WEIGHTED GRAPH

Motivated by considerations of the quadratic orthogonal bisectional curvature, we address the question of when a weighted graph (with possibly negative weights) has nonnegative Dirichlet energy.

On the Topology of Warped Product Pointwise Semi-Slant Submanifolds with Positive Curvature

In this paper, we obtain some topological characterizations for the warping function of a warped product pointwise semi-slant submanifold of the form Ωn=NTl×fNϕk in a complex projective space CP2m(4). Additionally, we will find certain restrictions on the warping function f, Dirichlet energy function E(f), and first non-zero eigenvalue λ1 to prove that stable l-currents do not exist and also that the homology groups have vanished in Ωn. As an application of the non-existence of the stable currents in Ωn, we show that the fundamental group π1(Ωn) is trivial and Ωn is simply connected under the same extrinsic conditions. Further, some similar conclusions are provided for CR-warped product submanifolds.

Energy Minimisers with Prescribed Jacobian

We consider the class of planar maps with Jacobian prescribed to be a fixed radially symmetric function f and which, moreover, fixes the boundary of a ball; we then study maps which minimise the 2p-Dirichlet energy in this class. We find a quantity $$\lambda [f]$$ λ [ f ] which controls the symmetry, uniqueness and regularity of minimisers: if $$\lambda [f]\le 1$$ λ [ f ] ≤ 1 then minimisers are symmetric and unique; if $$\lambda [f]$$ λ [ f ] is large but finite then there may be uncountably many minimisers, none of which is symmetric, although all of them have optimal regularity; if $$\lambda [f]$$ λ [ f ] is infinite then generically minimisers have lower regularity. In particular, this result gives a negative answer to a question of Hélein (Ann. Inst. H. Poincaré Anal. Non Linéaire 11(3):275–296, 1994). Some of our results also extend to the setting where the ball is replaced by $${\mathbb {R}}^2$$ R 2 and boundary conditions are not prescribed.

Geometry of Bi-Warped Product Submanifolds of Nearly Trans-Sasakian Manifolds

In the present work, we consider two types of bi-warped product submanifolds, M=MT×f1M⊥×f2Mϕ and M=Mϕ×f1MT×f2M⊥, in nearly trans-Sasakian manifolds and construct inequalities for the squared norm of the second fundamental form. The main results here are a generalization of several previous results. We also design some applications, in view of mathematical physics, and obtain relations between the second fundamental form and the Dirichlet energy. The relationship between the eigenvalues and the second fundamental form is also established.

A continuously perturbed Dirichlet energy with area-preserving stationary points that ‘buckle’ and occur in equal-energy pairs

We exhibit a family of convex functionals with infinitely many equal-energy $$C^1$$ C 1 stationary points that (i) occur in pairs $$v_{\pm }$$ v ± satisfying $$\det \nabla v_{\pm }=1$$ det ∇ v ± = 1 on the unit ball B in $${\mathbb {R}}^2$$ R 2 and (ii) obey the boundary condition $$v_{\pm }=\text {id}$$ v ± = id on $$ \partial B$$ ∂ B . When the parameter $$\epsilon $$ ϵ upon which the family of functionals depends exceeds $$\sqrt{2}$$ 2 , the stationary points appear to ‘buckle’ near the centre of B and their energies increase monotonically with the amount of buckling to which B is subjected. We also find Lagrange multipliers associated with the maps $$v_{\pm }(x)$$ v ± ( x ) and prove that they are proportional to $$(\epsilon -1/\epsilon )\ln |x|$$ ( ϵ - 1 / ϵ ) ln | x | as $$x \rightarrow 0$$ x → 0 in B. The lowest-energy pairs $$v_{\pm }$$ v ± are energy minimizers within the class of twist maps (see Taheri in Topol Methods Nonlinear Anal 33(1):179–204, 2009 or Sivaloganathan and Spector in Arch Ration Mech Anal 196:363–394, 2010), which, for each $$0\le r\le 1$$ 0 ≤ r ≤ 1 , take the circle $$\{x\in B: \ |x|=r\}$$ { x ∈ B : | x | = r } to itself; a fortiori, all $$v_{\pm }$$ v ± are stationary in the class of $$W^{1,2}(B;{\mathbb {R}}^2)$$ W 1 , 2 ( B ; R 2 ) maps w obeying $$w=\text {id}$$ w = id on $$\partial B$$ ∂ B and $$\det \nabla w=1$$ det ∇ w = 1 in B.

Geometric Inequalities of Bi-Warped Product Submanifolds of Nearly Kenmotsu Manifolds and Their Applications

The present paper aims to construct an inequality for bi-warped product submanifolds in a special class of almost metric manifolds, namely nearly Kenmotsu manifolds. As geometric applications, some exceptional cases that generalized several other inequalities are discussed. We also deliberate some applications in the context of mathematical physics and derive a new relation between the Dirichlet energy and the second fundamental form. Finally, we present a constructive remark at the end of this paper which shows the motive of the study.