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Let $X$ be either a general hypersurface of degree $n+1$ in $\mathbb P^n$ ora general $(2,n)$ complete intersection in $\mathbb P^{n+1}, n\geq 4$. Weconstruct balanced rational curves on $X$ of all high enough degrees. If $n=3$or $g=1$, we construct rigid curves of genus $g$ on $X$ of all high enoughdegrees. As an application we construct some rigid bundles on Calabi-Yauthreefolds. In addition, we construct some low-degree balanced rational curveson hypersurfaces of degree $n + 2$ in $\mathbb P^n$.

Regular curves on some Calabi-Yau and general-type complete intersections