# Bending the hemisphere

Let $$S^+$$ be the upper hemisphere of the standard sphere in $$\mathbb R^3$$ and $$b$$ -- the boundary of $$S^+$$ (the equator). Let $$b'$$ be a small isometric deformation of $$b$$ (a nearby curve of the same length). Is there a smooth (or of regularity higher than $$C^2$$) isometric immersion of $$S^+$$ which bounds $$b'$$?

I'm also interested in the infinitesimal version, i.e. if $$X$$ is a vector field along $$b$$ that does not stretch $$b$$, can $$X$$ be extended to a vector field along $$S^+$$ that is the derivative of a family of isometric immersions of $$S^+$$?

In other words: can an open set in the space of configurations of $$b$$ coincide with the subspace of configurations of $$b$$ induced from isometric immersions of $$S^+$$?

One can parametrize the upper hemisphere by the unit disk $$x^2+y^2\le 1$$ conformally by the well-known formula $$F(x,y) = \left(\frac{2x}{1+x^2+y^2},\frac{2y}{1+x^2+y^2},\frac{1-x^2-y^2}{1+x^2+y^2}\right).$$ Let $$V(x,y)$$ be a vector field along the image of $$F$$, which can be expressed uniquely in the form $$V(x,y) = a(x,y)\,F_x(x,y) + b(x,y)\,F_y(x,y) + c(x,y)\,F(x,y)$$ for some functions $$a$$, $$b$$, and $$c$$ on the unit disk. Then the condition that $$V$$ determine an infinitesimal isometric deformation, i.e., $$\mathrm{d}F \cdot \mathrm{d}V = 0$$, is easily seen to be the system of $$3$$ equations $$a_x-b_y = 0,\qquad a_y+b_x = 0,\qquad c = \frac{2(xa+yb)}{1+x^2+y^2} - a_x\,.$$ Thus, $$a+ib$$ must be a holomorphic function of $$z = x+iy$$, and $$c$$ is determined in explicitly in terms of $$a$$ and $$b$$. In particular, $$a + ib = \sum_{n=0}^{\infty} c_n\,z^n$$ for some complex coefficients $$c_n$$, $$n\ge 0$$.
Now, restrict everything to the boundary of the disk, set $$E_0(\theta) = F(\cos\theta,\sin\theta),\qquad E_1(\theta) = F_x(\cos\theta,\sin\theta),\qquad E_2(\theta) = F_y(\cos\theta,\sin\theta),$$ so that $$E_0$$, $$E_1$$, and $$E_2$$ are an orthonormal frame field along the boundary curve (i.e., the equator). Let $$W(\theta)$$ be a vector field along the boundary. It can be written uniquely in the form $$W(\theta) = f_0(\theta)\,E_0(\theta)+f_1(\theta)\,E_1(\theta)+f_2(\theta)\,E_2(\theta)$$ for some $$2\pi$$-periodic functions of $$\theta$$. The condition that $$W$$ furnish an infinitesimal isometric deformation of the boundary curve, i.e., $$\mathrm{d}E_0\cdot\mathrm{d}W = 0$$, is easily seen to be the differential equation $$f_0(\theta) = \frac{\mathrm{d}}{\mathrm{d}\theta}\bigl(\sin\theta\,f_1(\theta)-\cos\theta\,f_2(\theta)\bigr)$$ Notice that $$f_1$$ and $$f_2$$ can be arbitrary $$2\pi$$-periodic functions of $$\theta$$. However, if $$W$$ is to be the boundary value of an isometric deformation vector field $$V$$ as above, we will have to have $$f_1 + i f_2 = a(\cos\theta,\sin\theta) + ib(\cos\theta,\sin\theta) = \sum_{n=0}^{\infty} c_n\,\mathrm{e}^{in\theta},$$ i.e., the 'negative' Fourier coefficients of $$f_1+if_2$$ must all vanish.