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Let $\mathbb {D}\subset \mathbb {C}$ be the unit disk and let $\gamma_{1},\gamma _{2}:[0,T]\to\overline{\mathbb {D}}\setminus\{0\}$ be parametrizations of two slits $\Gamma_{1}:=\gamma(0,T], \Gamma_{2}:=\gamma_{2}(0,T]$ such that $\Gamma_{1}$ and $\Gamma_{2}$ are disjoint.

In this paper, we establish an analytic foundation for a fully non-linear equation 2 1 = f on manifolds with metrics of positive scalar curvature and apply it to give a (rough) classification of such manifolds. A crucial point is a simple observation that this equation is a degenerate elliptic equation without any condition on the sign of f and it is elliptic not only for f > 0 but also for f < 0. By defining a Yamabe constant Y2,1 with respect to this equation, we show that a manifold with metrics of positive scalar curvature admits a conformal metric of positive scalar curvature and positive 2-scalar curvature if and only if Y2,1 > 0. We give a complete solution for the corresponding Yamabe problem. Namely, let g0 be a positive scalar curvature metric, then in its conformal class there is a conformal metric with 2(g) = 1(g), for some constant . Using these analytic results, we give a rough classification of the space of manifolds with metrics of positive scalar curvature.

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Morphing is the process of changing a geometric model or an image into another. The process generally involves rigid body motions and non-rigid deformations. It is well known that there exists a unique conformal mapping from a simply connected surface into a unit disk by the Riemann mapping theorem. On the other hand, a 3D surface deformable model can be built via various approaches such as mutual parameterization from direct interpolation or surface matching using landmarks. In this paper, a numerical methods of 3D surface morphing based on deformable model and conformal mapping is demonstrated. We take the advantage of the unique representation of 3D surfaces by the mean curvatures $H$ and the conformal factors $\lambda$ associated with the Riemann mapping and build up the deformation model by consistently registering the landmarks on the conformal parametric domains. As a result, the correspondence of the $(H, \lambda)$ between two surfaces can be defined and a 3D deformation field can be reconstructed. Furthermore, by composition of the M\"obius transformation and the 3D deformation field, a smooth morphing sequence can be generated over a consistent mesh structure via the cubic spline homotopy. Several numerical experiments on the face morphing are presented to demonstrate the robustness of our approach.