By Qifan Y.
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15: One strictly rising and two strictly falling paths connecting y, u, and v to the boundary. 15. In either case, we get a piece of the triangulation bounded by vertices with non-positive function values. Other than u and v all other vertices in this boundary have strictly negative function values. If z belongs to the boundary of this piece then it has strictly negative function value simply because it differs from u and v. Else it belongs to the interior of the piece and we have h(f (z)) < 0 by the maximum principle.
Second, letting yuv and zuv be the two triangles sharing the interior edge uv in G, the points f (y) and f (z) lie on opposite sides of the line h−1 (0) that passes through f (u) and f (v). To see this, assume h(f (y)) > 0 and find a strictly rising path connecting y to the boundary. It exists because h(f (y)) > h(f (u)) so one of the neighbors of y has strictly larger function value, and the same is true for the next vertex on the path and so on. 15. 15: One strictly rising and two strictly falling paths connecting y, u, and v to the boundary.
Immersions of the Klein bottle. 2. The surface in that drawing intersects itself along a 50 II Surfaces path which ends at two branch points. In the smooth case, we get rankdeficient Jacobians at the branch points implying that this is not the image of an immersion. However, the Klein bottle can also be mapped without branch points and we conclude this section with the description of two such mappings. 12: Two immersions of the Klein bottle. Both models intersect themselves in a closed curve whose preimage are two loops.